{"id":991,"date":"2026-10-06T21:32:01","date_gmt":"2026-10-06T21:32:01","guid":{"rendered":"https:\/\/academic.pargaidefenceacademy.in\/Class-9th\/?p=991"},"modified":"2026-10-06T21:50:06","modified_gmt":"2026-10-06T21:50:06","slug":"8-%e0%a4%85%e0%a4%97%e0%a4%b2%e0%a5%87-%e0%a4%aa%e0%a4%a6-%e0%a4%95%e0%a4%be-%e0%a4%aa%e0%a5%82%e0%a4%b0%e0%a5%8d%e0%a4%b5%e0%a4%be%e0%a4%a8%e0%a5%81%e0%a4%ae%e0%a4%be%e0%a4%a8-%e0%a4%85","status":"publish","type":"post","link":"https:\/\/academic.pargaidefenceacademy.in\/Class-9th\/2026\/10\/06\/8-%e0%a4%85%e0%a4%97%e0%a4%b2%e0%a5%87-%e0%a4%aa%e0%a4%a6-%e0%a4%95%e0%a4%be-%e0%a4%aa%e0%a5%82%e0%a4%b0%e0%a5%8d%e0%a4%b5%e0%a4%be%e0%a4%a8%e0%a5%81%e0%a4%ae%e0%a4%be%e0%a4%a8-%e0%a4%85\/","title":{"rendered":"*8* | \u0905\u0917\u0932\u0947 \u092a\u0926 \u0915\u093e \u092a\u0942\u0930\u094d\u0935\u093e\u0928\u0941\u092e\u093e\u0928 \u2014 \u0905\u0928\u0941\u0915\u094d\u0930\u092e\u094b\u0902 \u0914\u0930 \u0936\u094d\u0930\u0947\u0922\u093c\u093f\u092f\u094b\u0902 \u0915\u093e \u0905\u0928\u094d\u0935\u0947\u0937\u0923 | Predicting the Next Term \u2014 Exploration of Sequences and Series"},"content":{"rendered":"<div id=\"pl-991\"  class=\"panel-layout\" ><div id=\"pg-991-0\"  class=\"panel-grid panel-no-style\" ><div id=\"pgc-991-0-0\"  class=\"panel-grid-cell\" ><div id=\"panel-991-0-0-0\" class=\"widget_text so-panel widget widget_custom_html panel-first-child panel-last-child\" data-index=\"0\" ><div class=\"textwidget custom-html-widget\"><div id=\"CH8-SEQ-INTRO\" style=\"max-width:860px;margin:12px auto;border:4px solid #0f5e4b;border-radius:14px;background:#e6f4f1;font-family:'Noto Sans Devanagari',Arial,sans-serif;overflow:hidden;\">\n<div style=\"background:#0a5c4b;padding:10px;text-align:center;\"><span style=\"background:#fff;color:#0a3d33;padding:7px 20px;border-radius:20px;font-weight:bold;font-size:13px;\">\u0905\u0927\u094d\u092f\u093e\u092f 8 - \u0905\u0928\u0941\u0915\u094d\u0930\u092e \u0914\u0930 \u0936\u094d\u0930\u0947\u0922\u093c\u0940 - \u0938\u093e\u092e\u093e\u0928\u094d\u092f \u092a\u0930\u093f\u091a\u092f \u0936\u0941\u0926\u094d\u0927<\/span><\/div>\n\n<div style=\"padding:10px 12px;\">\n<div id=\"sq-1\" style=\"display:block;\"><div style=\"background:#d32f2f;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udd34 \u092d\u093e\u0917 1\/5 - \u0905\u0928\u0941\u0915\u094d\u0930\u092e \u0915\u094d\u092f\u093e \u0939\u0948?<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:14px;line-height:2;height:560px;overflow-y:auto;\">\n\n<div style=\"background:#ffebee;border-left:4px solid #d32f2f;padding:8px 10px;font-weight:bold;\">8 | \u0905\u0917\u0932\u0947 \u092a\u0926 \u0915\u093e \u092a\u0942\u0930\u094d\u0935\u093e\u0928\u0941\u092e\u093e\u0928 \u2014 \u0905\u0928\u0941\u0915\u094d\u0930\u092e\u094b\u0902 \u0914\u0930 \u0936\u094d\u0930\u0947\u0922\u093c\u093f\u092f\u094b\u0902 \u0915\u093e \u0905\u0928\u094d\u0935\u0947\u0937\u0923<\/div>\n\n<b>\u0905\u0928\u0941\u0915\u094d\u0930\u092e \u0915\u094d\u092f\u093e \u0939\u0948?<\/b><br>\n\u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0940 \u0910\u0938\u0940 \u0938\u0942\u091a\u0940 \u091c\u094b \u0915\u093f\u0938\u0940 \u0928\u093f\u0936\u094d\u091a\u093f\u0924 \u0928\u093f\u092f\u092e \u0915\u0947 \u0905\u0928\u0941\u0938\u093e\u0930 \u0935\u094d\u092f\u0935\u0938\u094d\u0925\u093f\u0924 \u0939\u094b, \u0905\u0928\u0941\u0915\u094d\u0930\u092e \u0915\u0939\u0932\u093e\u0924\u0940 \u0939\u0948\u0964<br>\n\u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u0938\u0902\u0916\u094d\u092f\u093e \u0915\u094b \u0905\u0928\u0941\u0915\u094d\u0930\u092e \u0915\u093e \u092a\u0926 \u0915\u0939\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964<br><br>\n\n<b>\u0909\u0926\u093e\u0939\u0930\u0923:<\/b><br>\n- 2,4,6,8,10,... \u0938\u092e \u0938\u0902\u0916\u094d\u092f\u093e\u090f\u0901 - \u0928\u093f\u092f\u092e +2<br>\n- 1,3,5,7,9,... \u0935\u093f\u0937\u092e \u0938\u0902\u0916\u094d\u092f\u093e\u090f\u0901 - \u0928\u093f\u092f\u092e +2<br>\n- 1,4,9,16,25,... \u0935\u0930\u094d\u0917 \u0938\u0902\u0916\u094d\u092f\u093e\u090f\u0901 - n\u00b2<br>\n- 2,4,8,16,32,... \u0926\u094b\u0917\u0941\u0928\u093e - \u00d72<br><br>\n\n<div style=\"background:#fff9c4;border:2px solid #fbc02d;padding:8px;border-radius:8px;text-align:center;font-weight:bold;\">\n\u0905\u0928\u0941\u0915\u094d\u0930\u092e: a1, a2, a3, a4, ... an ...<br>\na1 \u092a\u0939\u0932\u093e \u092a\u0926, an n\u0935\u093e\u0901 \u092a\u0926 (\u0935\u094d\u092f\u093e\u092a\u0915 \u092a\u0926)<br>\n\u0928\u093f\u092f\u092e \u0938\u0947 \u0905\u0917\u0932\u093e \u092a\u0926 \u091c\u094d\u091e\u093e\u0924 \u0915\u0930\u0928\u093e = \u092a\u0942\u0930\u094d\u0935\u093e\u0928\u0941\u092e\u093e\u0928\n<\/div><br>\n\n<b>\u0936\u094d\u0930\u0947\u0922\u093c\u0940 \u0915\u094d\u092f\u093e \u0939\u0948?<\/b><br>\n\u0905\u0928\u0941\u0915\u094d\u0930\u092e \u0915\u0947 \u092a\u0926\u094b\u0902 \u0915\u093e \u092f\u094b\u0917 \u0936\u094d\u0930\u0947\u0922\u093c\u0940 \u0915\u0939\u0932\u093e\u0924\u093e \u0939\u0948\u0964<br>\n2+4+6+8+... \u0905\u0928\u0941\u0915\u094d\u0930\u092e 2,4,6,8 \u0915\u0940 \u0936\u094d\u0930\u0947\u0922\u093c\u0940<br><br>\n\n\u0905\u0917\u094d\u0928\u093f\u0935\u0940\u0930 \u092e\u0947\u0902 \u0905\u0928\u0941\u0915\u094d\u0930\u092e \u0938\u0947 1-2 \u092a\u094d\u0930\u0936\u094d\u0928 - \u0905\u0917\u0932\u093e \u092a\u0926, n\u0935\u093e\u0901 \u092a\u0926, \u092f\u094b\u0917<br>\n\n<\/div><\/div>\n\n<div id=\"sq-2\" style=\"display:none;\"><div style=\"background:#1565c0;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udd35 \u092d\u093e\u0917 2\/5 - \u0905\u0928\u0941\u0915\u094d\u0930\u092e \u0915\u0947 \u092a\u094d\u0930\u0915\u093e\u0930<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:14px;line-height:1.9;height:560px;overflow-y:auto;\">\n\n<b>\u0905\u0928\u0941\u0915\u094d\u0930\u092e \u0915\u0947 \u092a\u094d\u0930\u0915\u093e\u0930:<\/b><br><br>\n\n<b>1. \u0938\u092e\u093e\u0928\u094d\u0924\u0930 \u0936\u094d\u0930\u0947\u0922\u093c\u0940 (AP - Arithmetic Progression):<\/b><br>\n\u091c\u0939\u093e\u0901 \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u092a\u0926 \u092e\u0947\u0902 \u0938\u092e\u093e\u0928 \u0905\u0902\u0924\u0930 d \u091c\u0941\u0921\u093c\u0924\u093e \u0939\u0948\u0964<br>\na, a+d, a+2d, a+3d, ...<br>\n\u0909\u0926\u093e\u0939\u0930\u0923: 3,7,11,15,... d=4<br>\n\u0938\u0942\u0924\u094d\u0930: an = a+(n-1)d<br>\n\u092f\u094b\u0917 Sn = n\/2[2a+(n-1)d] = n\/2(a+an)<br><br>\n\n<b>2. \u0917\u0941\u0923\u094b\u0924\u094d\u0924\u0930 \u0936\u094d\u0930\u0947\u0922\u093c\u0940 (GP - Geometric Progression):<\/b><br>\n\u091c\u0939\u093e\u0901 \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u092a\u0926 \u092e\u0947\u0902 \u0938\u092e\u093e\u0928 \u0905\u0928\u0941\u092a\u093e\u0924 r \u0938\u0947 \u0917\u0941\u0923\u093e \u0939\u094b\u0924\u093e \u0939\u0948\u0964<br>\na, ar, ar\u00b2, ar\u00b3, ...<br>\n\u0909\u0926\u093e\u0939\u0930\u0923: 2,6,18,54,... r=3<br>\n\u0938\u0942\u0924\u094d\u0930: an = a\u00b7r^(n-1)<br>\n\u092f\u094b\u0917 Sn = a(r^n-1)\/(r-1), r\u22601<br><br>\n\n<b>3. \u0935\u093f\u0936\u0947\u0937 \u0905\u0928\u0941\u0915\u094d\u0930\u092e:<\/b><br>\n- \u0935\u0930\u094d\u0917: 1,4,9,16... n\u00b2<br>\n- \u0918\u0928: 1,8,27,64... n\u00b3<br>\n- \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u0940\u092f: 1,3,6,10... n(n+1)\/2<br>\n- \u092b\u093f\u092c\u094b\u0928\u093e\u091a\u0940: 1,1,2,3,5,8... an=an-1+an-2<br>\n\n<\/div><\/div>\n\n<div id=\"sq-3\" style=\"display:none;\"><div style=\"background:#2e7d32;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udfe2 \u092d\u093e\u0917 3\/5 - \u0905\u0917\u0932\u093e \u092a\u0926 \u0915\u0948\u0938\u0947 \u091c\u094d\u091e\u093e\u0924 \u0915\u0930\u0947\u0902?<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:14px;line-height:1.9;height:560px;overflow-y:auto;\">\n\n<b>\u0905\u0917\u0932\u0947 \u092a\u0926 \u0915\u093e \u092a\u0942\u0930\u094d\u0935\u093e\u0928\u0941\u092e\u093e\u0928 - \u092a\u0948\u091f\u0930\u094d\u0928 \u0926\u0947\u0916\u094b:<\/b><br><br>\n\n<b>\u0935\u093f\u0927\u093f 1 - \u0905\u0902\u0924\u0930 \u0926\u0947\u0916\u094b (AP):<\/b><br>\n2,5,8,11,... \u0905\u0902\u0924\u0930 3, \u0905\u0917\u0932\u093e 14<br>\nan=3n-1<br><br>\n\n<b>\u0935\u093f\u0927\u093f 2 - \u0905\u0928\u0941\u092a\u093e\u0924 \u0926\u0947\u0916\u094b (GP):<\/b><br>\n3,6,12,24,... \u0905\u0928\u0941\u092a\u093e\u0924 2, \u0905\u0917\u0932\u093e 48<br>\nan=3\u00b72^(n-1)<br><br>\n\n<b>\u0935\u093f\u0927\u093f 3 - \u0935\u0930\u094d\u0917\/\u0918\u0928:<\/b><br>\n1,4,9,16,... n\u00b2 \u0905\u0917\u0932\u093e 25<br>\n1,8,27,64,... n\u00b3 \u0905\u0917\u0932\u093e 125<br><br>\n\n<b>\u0935\u093f\u0927\u093f 4 - \u0926\u094b \u0928\u093f\u092f\u092e \u092e\u093f\u0932\u093e\u0915\u0930:<\/b><br>\n2,6,12,20,30,... n(n+1) \u0905\u0917\u0932\u093e 42<br>\n1,2,4,7,11,... \u0905\u0902\u0924\u0930 1,2,3,4 \u0905\u0917\u0932\u093e 16<br><br>\n\n<div style=\"background:#e8f5e9;border:2px solid #2e7d32;padding:8px;border-radius:8px;text-align:center;font-weight:bold;\">\n\u091f\u094d\u0930\u093f\u0915 \u0905\u0917\u094d\u0928\u093f\u0935\u0940\u0930:<br>\n\u0905\u0902\u0924\u0930 \u0938\u092e\u093e\u0928 \u2192 AP<br>\n\u0905\u0928\u0941\u092a\u093e\u0924 \u0938\u092e\u093e\u0928 \u2192 GP<br>\n\u0905\u0902\u0924\u0930 \u0915\u093e \u0905\u0902\u0924\u0930 \u0938\u092e\u093e\u0928 \u2192 \u0935\u0930\u094d\u0917\u0940\u092f<br>\n\u0928\u0939\u0940\u0902 \u0924\u094b n\u00b2\u00b1, n\u00b3\u00b1, \u00d72\u00b1 \u0926\u0947\u0916\u094b\n<\/div>\n\n<\/div><\/div>\n\n<div id=\"sq-4\" style=\"display:none;\"><div style=\"background:#6a1b9a;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udfe3 \u092d\u093e\u0917 4\/5 - \u0936\u094d\u0930\u0947\u0922\u093c\u0940 \u0915\u093e \u092f\u094b\u0917<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:14px;line-height:1.9;height:560px;overflow-y:auto;\">\n\n<b>\u0936\u094d\u0930\u0947\u0922\u093c\u0940 \u092f\u094b\u0917 \u0915\u0947 \u0938\u0942\u0924\u094d\u0930 \u0936\u0941\u0926\u094d\u0927:<\/b><br><br>\n\n<div style=\"background:#fff9c4;border:2px solid #fbc02d;padding:8px;border-radius:8px;\">\n<b>AP \u092f\u094b\u0917:<\/b><br>\nSn = n\/2[2a+(n-1)d]\n\u092f\u093e Sn = n\/2(a+l), l \u0905\u0902\u0924\u093f\u092e \u092a\u0926<br>\n\u0909\u0926\u093e\u0939\u0930\u0923: 1+2+3+...+100 =100\/2(1+100)=5050<br><br>\n\n<b>GP \u092f\u094b\u0917:<\/b><br>\nSn = a(r^n-1)\/(r-1) \u092f\u0926\u093f r>1<br>\nSn = a(1-r^n)\/(1-r) \u092f\u0926\u093f r<1<br>\n\u0905\u0928\u0902\u0924 GP |r|<1 \u0924\u094b S\u221e=a\/(1-r)<br>\n\u0909\u0926\u093e\u0939\u0930\u0923: 1+1\/2+1\/4+...=1\/(1-1\/2)=2<br><br>\n\n<b>\u0935\u093f\u0936\u0947\u0937 \u092f\u094b\u0917:<\/b><br>\n1+2+...+n = n(n+1)\/2<br>\n1\u00b2+2\u00b2+...+n\u00b2 = n(n+1)(2n+1)\/6<br>\n1\u00b3+2\u00b3+...+n\u00b3 = [n(n+1)\/2]\u00b2\n<\/div><br>\n\n\u0905\u0917\u094d\u0928\u093f\u0935\u0940\u0930 \u092e\u0947\u0902 \u092f\u094b\u0917 \u0935\u093e\u0932\u0947 \u092a\u094d\u0930\u0936\u094d\u0928 1-2 \u0906\u0924\u0947 \u0939\u0948\u0902 - n=100 \u0924\u0915 \u092f\u094b\u0917, AP \u0915\u093e 10 \u092a\u0926\u094b\u0902 \u0915\u093e \u092f\u094b\u0917<br>\n\n<\/div><\/div>\n\n<div id=\"sq-5\" style=\"display:none;\"><div style=\"background:#ff6f00;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udfe0 \u092d\u093e\u0917 5\/5 - \u0938\u093e\u0930\u093e\u0902\u0936<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:14px;line-height:2;height:560px;overflow-y:auto;\">\n\n<div style=\"background:#e3f2fd;border:2px solid #1565c0;padding:10px;border-radius:10px;text-align:center;font-weight:bold;\">\n<b>\u0905\u0927\u094d\u092f\u093e\u092f 8 \u0938\u093e\u0930\u093e\u0902\u0936 - \u0938\u093e\u092e\u093e\u0928\u094d\u092f \u092a\u0930\u093f\u091a\u092f<\/b><br>\n\u0905\u0928\u0941\u0915\u094d\u0930\u092e = \u0928\u093f\u092f\u092e \u0938\u0947 \u0935\u094d\u092f\u0935\u0938\u094d\u0925\u093f\u0924 \u0938\u0942\u091a\u0940 a1,a2,a3...<br>\n\u0936\u094d\u0930\u0947\u0922\u093c\u0940 = \u0905\u0928\u0941\u0915\u094d\u0930\u092e \u0915\u093e \u092f\u094b\u0917<br>\nAP: a,a+d,a+2d... an=a+(n-1)d, Sn=n\/2[2a+(n-1)d]\nGP: a,ar,ar\u00b2... an=a\u00b7r^(n-1), Sn=a(r^n-1)\/(r-1)<br>\n\u0905\u0917\u0932\u093e \u092a\u0926 = \u092a\u0948\u091f\u0930\u094d\u0928 \u092a\u0939\u091a\u093e\u0928\u094b \u0905\u0902\u0924\u0930\/\u0905\u0928\u0941\u092a\u093e\u0924\/\u0935\u0930\u094d\u0917\/\u0918\u0928<br>\n\u092f\u094b\u0917: 1..n=n(n+1)\/2, \u0935\u0930\u094d\u0917 n(n+1)(2n+1)\/6\n<\/div><br>\n\n<b>\u0938\u094b\u091a\u093f\u090f:<\/b><br>\n\u091c\u0940\u0935\u0928 \u092e\u0947\u0902 \u0905\u0928\u0941\u0915\u094d\u0930\u092e \u0915\u0939\u093e\u0901?<br>\n- \u091c\u0928\u0938\u0902\u0916\u094d\u092f\u093e \u0935\u0943\u0926\u094d\u0927\u093f GP, \u092c\u094d\u092f\u093e\u091c GP, \u0938\u0940\u0922\u093c\u093f\u092f\u094b\u0902 \u0915\u0940 \u090a\u0901\u091a\u093e\u0908 AP, \u092b\u0942\u0932\u094b\u0902 \u0915\u0940 \u092a\u0902\u0916\u0941\u0921\u093c\u093f\u092f\u093e\u0901 \u092b\u093f\u092c\u094b\u0928\u093e\u091a\u0940<br><br>\n\n\u0905\u0917\u0932\u093e: 8.1 \u0938\u092e\u093e\u0928\u094d\u0924\u0930 \u0936\u094d\u0930\u0947\u0922\u093c\u0940, 8.2 \u0917\u0941\u0923\u094b\u0924\u094d\u0924\u0930 \u0936\u094d\u0930\u0947\u0922\u093c\u0940, \u092a\u094d\u0930\u0936\u094d\u0928\u093e\u0935\u0932\u0940<br>\n\n<div style=\"background:#fff9c4;border:3px solid #fbc02d;padding:10px;border-radius:10px;text-align:center;margin-top:10px;font-weight:bold;\">\n\u2705 \u0905\u0927\u094d\u092f\u093e\u092f 8 \u0938\u093e\u092e\u093e\u0928\u094d\u092f \u092a\u0930\u093f\u091a\u092f \u0936\u0941\u0926\u094d\u0927 \u0916\u0924\u094d\u092e<br>\n\u0905\u0928\u0941\u0915\u094d\u0930\u092e = \u0938\u0942\u091a\u0940, \u0936\u094d\u0930\u0947\u0922\u093c\u0940 = \u092f\u094b\u0917<br>\nAP \u0905\u0902\u0924\u0930 \u0938\u092e\u093e\u0928 d, GP \u0905\u0928\u0941\u092a\u093e\u0924 \u0938\u092e\u093e\u0928 r<br>\n\u0905\u0917\u0932\u093e \u092a\u0926 \u092a\u0948\u091f\u0930\u094d\u0928 \u0938\u0947<br>\n\u0905\u0917\u094d\u0928\u093f\u0935\u0940\u0930 \u092e\u0947\u0902 1-2 MCQ \u0907\u0938\u0940 \u0938\u0947\n<\/div>\n\n<\/div><\/div>\n<\/div>\n\n<div style=\"background:#0a5c4b;padding:12px;display:flex;justify-content:space-between;align-items:center;\">\n<button type=\"button\" id=\"sqp\" style=\"background:#ffcc00;color:#000;border:3px solid #fff;border-radius:30px;padding:10px 22px;font-weight:bold;cursor:pointer;min-width:100px;\">&larr; \u092a\u0940\u091b\u0947<\/button>\n<span id=\"sqi\" style=\"background:#fff;color:#0a5c4b;padding:8px 20px;border-radius:20px;font-weight:bold;\">1 \/ 5<\/span>\n<button type=\"button\" id=\"sqn\" style=\"background:#00e676;color:#000;border:3px solid #fff;border-radius:30px;padding:10px 22px;font-weight:bold;cursor:pointer;min-width:100px;\">\u0906\u0917\u0947 &rarr;<\/button>\n<\/div>\n<\/div>\n\n<script>\n(function(){\nvar c=1,t=5;\nfunction s(n){\nfor(var i=1;i<=t;i++){\nvar el=document.getElementById('sq-'+i);\nif(el)el.style.display=(i===n?'block':'none');\n}\ndocument.getElementById('sqi').textContent=n+' \/ '+t;\nc=n;\n}\ndocument.getElementById('sqn').onclick=function(){var n=c+1;if(n>t)n=1;s(n);};\ndocument.getElementById('sqp').onclick=function(){var n=c-1;if(n<1)n=t;s(n);};\n})();\n<\/script><\/div><\/div><\/div><\/div><div id=\"pg-991-1\"  class=\"panel-grid panel-no-style\" ><div id=\"pgc-991-1-0\"  class=\"panel-grid-cell\" ><div id=\"panel-991-1-0-0\" class=\"widget_text so-panel widget widget_custom_html panel-first-child panel-last-child\" data-index=\"1\" ><div class=\"textwidget custom-html-widget\"><div id=\"CH8-ALL\" style=\"max-width:860px;margin:12px auto;border:4px solid #0f5e4b;border-radius:14px;background:#e6f4f1;font-family:'Noto Sans Devanagari',Arial,sans-serif;overflow:hidden;\">\n<div style=\"background:#0a5c4b;padding:10px;text-align:center;\"><span style=\"background:#fff;color:#0a3d33;padding:7px 20px;border-radius:20px;font-weight:bold;font-size:13px;\">8.1-8.3 \u0905\u0928\u0941\u0915\u094d\u0930\u092e - \u0935\u094d\u092f\u093e\u092a\u0915 \u0914\u0930 \u092a\u0941\u0928\u0930\u093e\u0935\u0930\u094d\u0924\u0940 \u0938\u0942\u0924\u094d\u0930 - \u092a\u0947\u091c 175-179<\/span><\/div>\n\n<div style=\"padding:10px 12px;\">\n<div id=\"a8-1\" style=\"display:block;\"><div style=\"background:#d32f2f;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udd34 \u092d\u093e\u0917 1\/5 - 8.1 \u0905\u0928\u0941\u0915\u094d\u0930\u092e \u0938\u0947 \u092a\u0930\u093f\u091a\u092f<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:14px;line-height:2;height:560px;overflow-y:auto;\">\n\n<div style=\"background:#ffebee;border-left:4px solid #d32f2f;padding:8px 10px;font-weight:bold;\">8.1 \u0905\u0928\u0941\u0915\u094d\u0930\u092e \u0938\u0947 \u092a\u0930\u093f\u091a\u092f \u0936\u0941\u0926\u094d\u0927<\/div>\n\u0939\u092e \u0905\u092a\u0928\u0947 \u091a\u093e\u0930\u094b\u0902 \u0913\u0930 \u0938\u092d\u0940 \u0938\u094d\u0925\u093e\u0928\u094b\u0902 \u092a\u0930 \u0935\u093f\u092d\u093f\u0928\u094d\u0928 \u092a\u094d\u0930\u0915\u093e\u0930 \u0915\u0947 \u092a\u094d\u0930\u0924\u093f\u0930\u0942\u092a \u0905\u0930\u094d\u0925\u093e\u0924 \u092a\u0948\u091f\u0930\u094d\u0928 \u0926\u0947\u0916\u0924\u0947 \u0939\u0948\u0902, \u091a\u093e\u0939\u0947 \u0935\u0939 \u092a\u094d\u0930\u0915\u0943\u0924\u093f \u092e\u0947\u0902 \u0939\u094b, \u0915\u0932\u093e \u092e\u0947\u0902 \u0939\u094b, \u0938\u0902\u0917\u0940\u0924 \u092e\u0947\u0902 \u0939\u094b, \u0935\u093f\u0924\u094d\u0924 \u092e\u0947\u0902 \u0939\u094b \u0905\u0925\u0935\u093e \u0926\u0948\u0928\u093f\u0915 \u091c\u0940\u0935\u0928 \u0915\u0947 \u0915\u0908 \u0905\u0928\u094d\u092f \u0938\u0902\u0926\u0930\u094d\u092d\u094b\u0902 \u092e\u0947\u0902\u0964 \u092a\u094d\u0930\u0924\u093f\u0930\u0942\u092a, \u0935\u093f\u0936\u094d\u0935 \u0915\u094b \u0938\u092e\u091d\u0928\u0947 \u092e\u0947\u0902 \u0914\u0930 \u0906\u0917\u0947 \u0915\u094d\u092f\u093e \u0906\u090f\u0917\u093e, \u092f\u0939 \u092a\u0942\u0930\u094d\u0935\u093e\u0928\u0941\u092e\u093e\u0928 \u0932\u0917\u093e\u0928\u0947 \u092e\u0947\u0902 \u0938\u0939\u093e\u092f\u0924\u093e \u0915\u0930\u0924\u0947 \u0939\u0948\u0902\u0964 \u0917\u0923\u093f\u0924 \u092e\u0947\u0902 \u0905\u0928\u0941\u0915\u094d\u0930\u092e, \u092a\u094d\u0930\u0924\u093f\u0930\u0942\u092a\u094b\u0902 \u0915\u0947 \u0935\u093f\u0936\u0947\u0937 \u092a\u094d\u0930\u0915\u093e\u0930 \u0939\u0948\u0902, \u091c\u094b \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u092f\u093e \u0905\u0928\u094d\u092f \u0935\u0938\u094d\u0924\u0941\u0913\u0902 \u0915\u094b \u090f\u0915 \u0935\u093f\u0936\u0947\u0937 \u0915\u094d\u0930\u092e \u092e\u0947\u0902 \u0935\u094d\u092f\u0935\u0938\u094d\u0925\u093f\u0924 \u0915\u0930\u0928\u0947 \u092a\u0930 \u092c\u0928\u0924\u0947 \u0939\u0948\u0902\u0964<br><br>\n\n<b>\u092a\u0930\u093f\u092d\u093e\u0937\u093e \u0936\u0941\u0926\u094d\u0927:<\/b><br>\n\u0905\u0928\u0941\u0915\u094d\u0930\u092e \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0940 \u090f\u0915 \u0915\u094d\u0930\u092e\u093f\u0915 \u0938\u0942\u091a\u0940 \u0939\u0948, \u091c\u0939\u093e\u0901 \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u0938\u0902\u0916\u094d\u092f\u093e \u0905\u0928\u0941\u0915\u094d\u0930\u092e \u0915\u093e \u090f\u0915 \u092a\u0926 \u0939\u0948\u0964<br>\n\u0924\u0940\u0928 \u092c\u093f\u0902\u0926\u0941 ... \u092f\u0939 \u0926\u0930\u094d\u0936\u093e\u0924\u0947 \u0939\u0948\u0902 \u0915\u093f \u0905\u0928\u0941\u0915\u094d\u0930\u092e \u0905\u092a\u0930\u093f\u092e\u093f\u0924 \u0930\u0942\u092a \u0938\u0947 \u091a\u0932\u0924\u093e \u0930\u0939\u0924\u093e \u0939\u0948\u0964<br><br>\n\n<b>\u0909\u0926\u093e\u0939\u0930\u0923 \u091c\u094b \u0906\u092a \u0915\u0915\u094d\u0937\u093e 6,7,8 \u092e\u0947\u0902 \u0926\u0947\u0916 \u091a\u0941\u0915\u0947 \u0939\u0948\u0902:<\/b><br>\n1,2,3,4,5,6,... \u092a\u094d\u0930\u093e\u0915\u0943\u0924 \u0938\u0902\u0916\u094d\u092f\u093e\u090f\u0901<br>\n1,3,5,7,9,11,... \u0935\u093f\u0937\u092e \u0938\u0902\u0916\u094d\u092f\u093e\u090f\u0901<br>\n1,3,6,10,15,21,... \u0924\u094d\u0930\u093f\u092d\u0941\u091c\u093e\u0915\u093e\u0930 \u0938\u0902\u0916\u094d\u092f\u093e\u090f\u0901<br>\n1,4,9,16,25,36,... \u0935\u0930\u094d\u0917 \u0938\u0902\u0916\u094d\u092f\u093e\u090f\u0901<br><br>\n\n\u0905\u0928\u0941\u0915\u094d\u0930\u092e \u092a\u0930\u093f\u092e\u093f\u0924 \u092f\u093e \u0905\u092a\u0930\u093f\u092e\u093f\u0924 \u0939\u094b \u0938\u0915\u0924\u0947 \u0939\u0948\u0902\u0964 \u090a\u092a\u0930 \u0926\u093f\u090f \u0917\u090f \u0905\u092a\u0930\u093f\u092e\u093f\u0924 \u0939\u0948\u0902, \u0915\u093f\u0902\u0924\u0941 6,12,24,48,96 \u090f\u0915 \u092a\u0930\u093f\u092e\u093f\u0924 \u0905\u0928\u0941\u0915\u094d\u0930\u092e \u0939\u0948, \u091c\u093f\u0938\u092e\u0947\u0902 5 \u092a\u0926 \u0939\u0948\u0902\u0964<br><br>\n\n<div style=\"background:#fff9c4;border:2px solid #fbc02d;padding:8px;border-radius:8px;\">\n<b>\u092a\u0948\u091f\u0930\u094d\u0928 \u0936\u0941\u0926\u094d\u0927:<\/b><br>\n\u092a\u094d\u0930\u093e\u0915\u0943\u0924 \u092e\u0947\u0902 \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u092a\u0926 \u092a\u093f\u091b\u0932\u0940 \u0938\u0947 1 \u0905\u0927\u093f\u0915<br>\n\u0935\u093f\u0937\u092e \u092e\u0947\u0902 \u0926\u094b \u0915\u094d\u0930\u092e\u093e\u0917\u0924 \u092a\u0926\u094b\u0902 \u092e\u0947\u0902 \u0905\u0902\u0924\u0930 2<br>\n\u0924\u094d\u0930\u093f\u092d\u0941\u091c\u093e\u0915\u093e\u0930 \u092e\u0947\u0902 \u0915\u094d\u0930\u092e\u093e\u0917\u0924 \u0905\u0902\u0924\u0930 2,3,4,5,6 \u0939\u0948<br>\n\u0924\u094d\u0930\u093f\u092d\u0941\u091c\u093e\u0915\u093e\u0930 = \u092a\u094d\u0930\u093e\u0915\u0943\u0924 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u093e \u092f\u094b\u0917: 1=1, 3=1+2, 6=1+2+3, 10=1+2+3+4<br>\nn\u0935\u0940\u0902 \u0924\u094d\u0930\u093f\u092d\u0941\u091c\u093e\u0915\u093e\u0930 = 1+2+...+n = n(n+1)\/2<br>\n\u0935\u0930\u094d\u0917 \u092e\u0947\u0902 \u0905\u0902\u0924\u0930 3,5,7,9,11 \u0914\u0930 1=1, 4=1+3, 9=1+3+5, 16=1+3+5+7<br>\n\u0935\u0930\u094d\u0917 = \u0915\u094d\u0930\u092e\u093f\u0915 \u0935\u093f\u0937\u092e \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u093e \u092f\u094b\u0917\n<\/div>\n\n<\/div><\/div>\n\n<div id=\"a8-2\" style=\"display:none;\"><div style=\"background:#1565c0;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udd35 \u092d\u093e\u0917 2\/5 - \u0938\u0902\u0915\u0947\u0924\u0928 \u0914\u0930 \u092f\u094b\u0917 \u0905\u0928\u0941\u0915\u094d\u0930\u092e<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:14px;line-height:1.9;height:560px;overflow-y:auto;\">\n\n<b>\u0938\u0902\u0915\u0947\u0924\u0928 \u0936\u0941\u0926\u094d\u0927:<\/b><br>\n\u092a\u0939\u0932\u093e \u092a\u0926 t1, \u0926\u0942\u0938\u0930\u093e t2, \u0924\u0940\u0938\u0930\u093e t3,... n\u0935\u093e\u0901 \u092a\u0926 tn<br>\n\u0935\u093f\u0937\u092e \u0915\u0947 \u0932\u093f\u090f t1=1, t2=3, t3=5, t4=7<br>\nt4=7 \u092c\u0924\u093e\u0924\u093e \u0939\u0948 \u0915\u093f \u091a\u094c\u0925\u0940 \u0938\u094d\u0925\u093f\u0924\u093f \u092a\u0930 \u092a\u0926 7 \u0939\u0948\u0964<br>\n\u090f\u0915 \u0938\u0947 \u0905\u0927\u093f\u0915 \u0905\u0928\u0941\u0915\u094d\u0930\u092e \u0939\u094b\u0902 \u0924\u094b t, s, u \u0915\u093e \u092a\u094d\u0930\u092f\u094b\u0917: s1,s2,s3..., u1,u2,u3...<br><br>\n\n<b>\u092a\u094d\u0930\u0936\u094d\u0928 \u0936\u0941\u0926\u094d\u0927:<\/b><br>\n\u0905\u0928\u0941\u0915\u094d\u0930\u092e 1,4,7,10,13,... \u092a\u0930 \u0935\u093f\u091a\u093e\u0930 \u0915\u0940\u091c\u093f\u090f\u0964 \u0915\u094d\u092f\u093e \u0906\u092a \u0906\u0917\u093e\u092e\u0940 \u091a\u093e\u0930 \u092a\u0926\u094b\u0902 \u0915\u093e \u092a\u0942\u0930\u094d\u0935\u093e\u0928\u0941\u092e\u093e\u0928 \u0932\u0917\u093e \u0938\u0915\u0924\u0947 \u0939\u0948\u0902? \u0915\u094d\u092f\u093e \u0906\u092a \u0907\u0938 \u0905\u0928\u0941\u0915\u094d\u0930\u092e \u092e\u0947\u0902 \u0926\u093f\u090f \u0917\u090f \u092a\u0926 \u0924\u0915 \u0906\u0928\u0947 \u0935\u093e\u0932\u0947 \u0938\u092d\u0940 \u092a\u0926\u094b\u0902 \u0915\u093e \u092f\u094b\u0917 \u0915\u0930\u0928\u0947 \u092a\u0930 \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0905\u0928\u0941\u0915\u094d\u0930\u092e \u0915\u0947 \u092a\u0939\u0932\u0947 10 \u092a\u0926 \u092c\u0928\u093e \u0938\u0915\u0924\u0947 \u0939\u0948\u0902?<br>\n\u0938\u0902\u0915\u0947\u0924 \u2014 \u092a\u0939\u0932\u093e \u092a\u0926 1 \u0939\u0948\u0964 \u0926\u0942\u0938\u0930\u093e \u092a\u0926 1+4=5 \u0939\u0948, \u0924\u0940\u0938\u0930\u093e \u092a\u0926 1+4+7=12 \u0939\u0948 \u0914\u0930 \u0906\u0917\u0947 \u092d\u0940 \u0907\u0938\u0940 \u092a\u094d\u0930\u0915\u093e\u0930 \u092a\u0926 \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0915\u093f\u090f \u091c\u093e \u0938\u0915\u0924\u0947 \u0939\u0948\u0902\u0964<br>\n\u0939\u0932: 1,4,7,10,13 \u0905\u0902\u0924\u0930 3 \u2192 \u0906\u0917\u093e\u092e\u0940 16,19,22,25<br>\n\u092f\u094b\u0917 \u0905\u0928\u0941\u0915\u094d\u0930\u092e: 1,5,12,22,35,51,70,92,117,145...<br><br>\n\n<b>\u0905\u0928\u0941\u0915\u094d\u0930\u092e \u0915\u0947 \u092a\u094d\u0930\u0915\u093e\u0930 \u0936\u0941\u0926\u094d\u0927:<\/b><br>\n- \u092a\u0926 \u092d\u093f\u0928\u094d\u0928, \u090b\u0923\u093e\u0924\u094d\u092e\u0915 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 \u092d\u0940 \u0939\u094b \u0938\u0915\u0924\u0947 \u0939\u0948\u0902<br>\n- \u092c\u0922\u093c\u0924\u0947: 1,2,3... \u0918\u091f\u0924\u0947: 1,1\/2,1\/3,1\/4...<br>\n- \u090b\u0923\u093e\u0924\u094d\u092e\u0915: -7,-3,1,5,9... S1=-7 \u0905\u0902\u0924\u0930 4<br>\n- tn \u0938\u0902\u0915\u0947\u0924 \u092e\u0947\u0902 n \u0938\u0926\u0948\u0935 \u092a\u094d\u0930\u093e\u0915\u0943\u0924 \u0938\u0902\u0916\u094d\u092f\u093e, \u092a\u0930 \u092a\u0926 \u0915\u094b\u0908 \u092d\u0940 \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e<br><br>\n\n<b>\u092a\u094d\u0930\u0936\u094d\u0928:<\/b> \u0924\u094d\u0930\u093f\u092d\u0941\u091c\u093e\u0915\u093e\u0930 \u0905\u0928\u0941\u0915\u094d\u0930\u092e \u0939\u0947\u0924\u0941 t5,t6,t7,t8 \u0932\u093f\u0916\u093f\u090f?<br>\n\u0909\u0924\u094d\u0924\u0930: t5=15, t6=21, t7=28, t8=36 (n(n+1)\/2)<br>\n\n<\/div><\/div>\n\n<div id=\"a8-3\" style=\"display:none;\"><div style=\"background:#2e7d32;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udfe2 \u092d\u093e\u0917 3\/5 - 8.2 \u0935\u094d\u092f\u093e\u092a\u0915 \u0938\u0942\u0924\u094d\u0930<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:14px;line-height:1.9;height:560px;overflow-y:auto;\">\n\n<div style=\"background:#e8f5e9;border-left:4px solid #2e7d32;padding:8px 10px;font-weight:bold;\">8.2 \u0905\u0928\u0941\u0915\u094d\u0930\u092e \u0915\u0947 \u0932\u093f\u090f \u0935\u094d\u092f\u093e\u092a\u0915 \u0938\u0942\u0924\u094d\u0930 \u0936\u0941\u0926\u094d\u0927<\/div>\n\u0938\u0902\u0915\u0947\u0924 tn \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0915\u0930\u0924\u0947 \u0939\u0941\u090f \u0939\u092e \u0915\u093f\u0938\u0940 \u0905\u0928\u0941\u0915\u094d\u0930\u092e \u0915\u0940 n\u0935\u0940\u0902 \u0938\u094d\u0925\u093f\u0924\u093f \u092a\u0930 \u0938\u094d\u0925\u093f\u0924 \u092a\u0926 \u0905\u0930\u094d\u0925\u093e\u0924 n\u0935\u0947\u0902 \u092a\u0926 \u0915\u0947 \u0932\u093f\u090f \u090f\u0915 \u0935\u094d\u092f\u093e\u092a\u0915 \u0938\u0942\u0924\u094d\u0930 \u0932\u093f\u0916 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902\u0964 \u0935\u094d\u092f\u093e\u092a\u0915 \u0938\u0942\u0924\u094d\u0930 \u0938\u0947 \u092a\u0926 \u0915\u093e \u092e\u093e\u0928 \u091c\u094d\u091e\u093e\u0924 \u0915\u0930\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u092a\u0926 \u0915\u0940 \u0938\u094d\u0925\u093f\u0924\u093f \u0938\u0902\u0916\u094d\u092f\u093e n \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964<br><br>\n\n<b>\u0909\u0926\u093e\u0939\u0930\u0923 1 \u0936\u0941\u0926\u094d\u0927:<\/b> \u0935\u094d\u092f\u0902\u091c\u0915 un=2n-1 \u092a\u0930 \u0935\u093f\u091a\u093e\u0930 \u0915\u0940\u091c\u093f\u090f\u0964 \u092f\u0939 \u0935\u094d\u092f\u0902\u091c\u0915 \u092c\u0924\u093e\u0924\u093e \u0939\u0948 \u0915\u093f \u0905\u0928\u0941\u0915\u094d\u0930\u092e \u0915\u093e n\u0935\u093e\u0901 \u092a\u0926 2n-1 \u0938\u0942\u0924\u094d\u0930 \u0926\u094d\u0935\u093e\u0930\u093e \u0926\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964 \u091c\u092c \u0939\u092e \u0935\u094d\u092f\u0902\u091c\u0915 2n-1 \u092e\u0947\u0902 n \u0915\u094b 1,2,3,... \u0938\u0947 \u0935\u093f\u0938\u094d\u0925\u093e\u092a\u093f\u0924 \u0915\u0930\u0924\u0947 \u0939\u0948\u0902, \u0924\u094b \u0939\u092e\u0947\u0902 u1=2\u00d71-1=1, u2=2\u00d72-1=3, u3=2\u00d73-1=5 \u0906\u0926\u093f \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0939\u094b\u0924\u0947 \u0939\u0948\u0902\u0964 \u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930, un=2n-1 \u0935\u093f\u0937\u092e \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0947 \u0905\u0928\u0941\u0915\u094d\u0930\u092e \u0915\u0947 n\u0935\u0947\u0902 \u092a\u0926 \u0915\u0947 \u0932\u093f\u090f \u0935\u094d\u092f\u093e\u092a\u0915 \u0938\u0942\u0924\u094d\u0930 \u0939\u0948\u0964<br><br>\n\n<div style=\"background:#fff9c4;border:2px solid #fbc02d;padding:8px;border-radius:8px;text-align:center;font-weight:bold;\">\n\u0935\u094d\u092f\u093e\u092a\u0915 \u0938\u0942\u0924\u094d\u0930 \u0915\u094d\u092f\u094b\u0902 \u0909\u092a\u092f\u094b\u0917\u0940?<br>\n20\u0935\u093e\u0901, 53\u0935\u093e\u0901, 300\u0935\u093e\u0901 \u092a\u0926 \u0938\u0940\u0927\u0947 n \u0930\u0916\u0915\u0930 \u091c\u094d\u091e\u093e\u0924 \u0915\u0930 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902, \u092c\u093f\u0928\u093e \u092a\u093f\u091b\u0932\u0947 \u092a\u0926 \u091c\u094d\u091e\u093e\u0924 \u0915\u093f\u090f\u0964<br>\n\u091c\u093e\u0901\u091a \u0938\u0915\u0924\u0947 \u0939\u0948\u0902 \u0915\u093f \u0915\u094b\u0908 \u0938\u0902\u0916\u094d\u092f\u093e \u092a\u0926 \u0939\u0948 \u092f\u093e \u0928\u0939\u0940\u0902\u0964\n<\/div><br>\n\n<b>53\u0935\u093e\u0901 \u0935\u093f\u0937\u092e:<\/b> u53=2\u00d753-1=105, u108=215, u1170=2339<br>\n<b>137 \u0915\u094c\u0928 \u0938\u0940 \u0938\u094d\u0925\u093f\u0924\u093f?<\/b> 2n-1=137 \u2192 n=69 \u2192 69\u0935\u093e\u0901 \u092a\u0926<br><br>\n\n<b>\u0909\u0926\u093e\u0939\u0930\u0923 2 \u0936\u0941\u0926\u094d\u0927:<\/b> sn=5n-2 \u0926\u094d\u0935\u093e\u0930\u093e \u092c\u0928\u093e\u092f\u093e \u0905\u0928\u0941\u0915\u094d\u0930\u092e\u0964 \u092a\u094d\u0930\u0925\u092e 6 \u092a\u0926 \u0932\u093f\u0916\u093f\u090f? 100\u0935\u093e\u0901? 1000\u0935\u093e\u0901? 308 \u0914\u0930 473 \u092a\u0926 \u0939\u0948\u0902?<br>\ns1=3, s2=8, s3=13, s4=18, s5=23, s6=28 (AP d=5)<br>\ns100=498, s1000=4998<br>\n308: 5n-2=308 \u2192 n=62 \u2192 62\u0935\u093e\u0901 \u092a\u0926 \u0939\u0948<br>\n471: 5n-2=471 \u2192 5n=473 \u2192 n=94.6 \u092a\u094d\u0930\u093e\u0915\u0943\u0924 \u0928\u0939\u0940\u0902 \u2192 \u092a\u0926 \u0928\u0939\u0940\u0902<br>\nn \u0915\u094b \u092a\u094d\u0930\u093e\u0915\u0943\u0924 \u0939\u094b\u0928\u093e \u091a\u093e\u0939\u093f\u090f \u0915\u094d\u092f\u094b\u0902\u0915\u093f \u092a\u0926-\u0938\u0902\u0916\u094d\u092f\u093e \u092a\u094d\u0930\u093e\u0915\u0943\u0924 \u0939\u094b\u0924\u0940 \u0939\u0948\u0964<br>\n\n<\/div><\/div>\n\n<div id=\"a8-4\" style=\"display:none;\"><div style=\"background:#6a1b9a;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udfe3 \u092d\u093e\u0917 4\/5 - 8.3 \u092a\u0941\u0928\u0930\u093e\u0935\u0930\u094d\u0924\u0940 \u0938\u0942\u0924\u094d\u0930<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:14px;line-height:1.9;height:560px;overflow-y:auto;\">\n\n<div style=\"background:#f3e5f5;border-left:4px solid #6a1b9a;padding:8px 10px;font-weight:bold;\">8.3 \u0905\u0928\u0941\u0915\u094d\u0930\u092e \u0915\u0947 \u0932\u093f\u090f \u092a\u0941\u0928\u0930\u093e\u0935\u0930\u094d\u0924\u0940 \u0938\u0942\u0924\u094d\u0930 \u0936\u0941\u0926\u094d\u0927<\/div>\n\u0905\u092c \u0924\u0915 \u0939\u092e\u0928\u0947 n\u0935\u0947\u0902 \u092a\u0926 \u0915\u0947 \u0932\u093f\u090f \u0935\u094d\u092f\u093e\u092a\u0915 \u0938\u0942\u0924\u094d\u0930 \u092e\u0947\u0902 n \u0915\u0947 \u092a\u0926\u094b\u0902 \u092e\u0947\u0902 \u090f\u0915 \u0938\u0942\u0924\u094d\u0930 \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0915\u093f\u092f\u093e \u0939\u0948\u0964 \u0905\u0928\u0941\u0915\u094d\u0930\u092e \u0915\u0947 \u0932\u093f\u090f \u0938\u0942\u0924\u094d\u0930 \u0932\u093f\u0916\u0928\u0947 \u0915\u0947 \u0905\u0928\u094d\u092f \u092a\u094d\u0930\u0915\u093e\u0930 \u092d\u0940 \u0939\u0948\u0902\u0964 \u0905\u0928\u0941\u0915\u094d\u0930\u092e 1,4,7,10,13,... \u092a\u0930 \u0935\u093f\u091a\u093e\u0930 \u0915\u0940\u091c\u093f\u090f\u0964 \u0907\u0938\u0915\u093e n\u0935\u093e\u0901 \u092a\u0926 tn=3n-2 \u0939\u0948\u0964 \u0927\u094d\u092f\u093e\u0928 \u0926\u0940\u091c\u093f\u090f \u0915\u093f \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u092a\u0926 \u0905\u092a\u0928\u0947 \u092a\u0942\u0930\u094d\u0935 \u092a\u0926 \u0938\u0947 3 \u0905\u0927\u093f\u0915 \u0939\u0948\u0964 t2=t1+3, t3=t2+3, t4=t3+3\u0964 \u0935\u094d\u092f\u093e\u092a\u0915 \u0930\u0942\u092a \u092e\u0947\u0902 tn=tn-1+3, \u091c\u0939\u093e\u0901 n\u22652 \u0939\u0948 \u0905\u0925\u0935\u093e t1=1, tn=tn-1+3\u0964 \u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930 \u0915\u093f\u0938\u0940 \u0905\u0928\u0941\u0915\u094d\u0930\u092e \u0915\u0947 \u092a\u0926\u094b\u0902 \u0915\u094b \u0909\u0928\u0938\u0947 \u092a\u0939\u0932\u0947 \u0906\u090f \u092a\u0926\u094b\u0902 \u0938\u0947 \u0938\u0902\u092c\u0926\u094d\u0927 \u0915\u0930\u0924\u0947 \u0939\u0941\u090f \u0905\u0928\u0941\u0915\u094d\u0930\u092e \u0915\u0940 \u0935\u094d\u092f\u093e\u0916\u094d\u092f\u093e \u0915\u0930\u0928\u093e \u092a\u0941\u0928\u0930\u093e\u0935\u0930\u094d\u0924\u0940 \u0938\u0942\u0924\u094d\u0930 \u0915\u0939\u0932\u093e\u0924\u093e \u0939\u0948\u0964<br><br>\n\n<b>\u0909\u0926\u093e\u0939\u0930\u0923 3 \u0936\u0941\u0926\u094d\u0927:<\/b> u1=1, un=2un-1+3, \u091c\u092c n\u22652 \u0939\u094b\u0964 \u092a\u094d\u0930\u0925\u092e \u091a\u093e\u0930 \u092a\u0926? \u0915\u094d\u092f\u093e 133 \u092a\u0926 \u0939\u0948?<br>\nu2=2\u00d71+3=5<br>\nu3=2\u00d75+3=13<br>\nu4=2\u00d713+3=29<br>\n\u092a\u094d\u0930\u0925\u092e \u091a\u093e\u0930 1,5,13,29\u0964 \u0906\u0917\u0947 u5=61, u6=125, u7=253 \u2192 133 \u092a\u0926 \u0928\u0939\u0940\u0902 (125 \u0914\u0930 253 \u0915\u0947 \u092c\u0940\u091a)<br><br>\n\n<b>\u0909\u0926\u093e\u0939\u0930\u0923 4 \u0936\u0941\u0926\u094d\u0927:<\/b> s1=3, sn=sn-1(sn-1-1), \u091c\u0939\u093e\u0901 n\u22652<br>\ns2=3\u00d72=6<br>\ns3=6\u00d75=30<br>\ns4=30\u00d729=870<br>\n\u092a\u094d\u0930\u0925\u092e \u091a\u093e\u0930 3,6,30,870<br><br>\n\n<b>\u0935\u093f\u0930\u0939\u093e\u0902\u0915-\u092b\u093f\u092c\u094b\u0928\u093e\u091a\u0940 \u0905\u0928\u0941\u0915\u094d\u0930\u092e \u0936\u0941\u0926\u094d\u0927:<\/b><br>\n\u0915\u093f\u0938\u0940 \u092d\u0940 \u092a\u0941\u0928\u0930\u093e\u0935\u0930\u094d\u0924\u0940 \u0938\u0942\u0924\u094d\u0930 \u092e\u0947\u0902 \u0915\u0947\u0935\u0932 \u090f\u0915 \u0939\u0940 \u092a\u0942\u0930\u094d\u0935 \u092a\u0926 \u0938\u092e\u094d\u092e\u093f\u0932\u093f\u0924 \u0928\u0939\u0940\u0902 \u0939\u094b\u0924\u093e, \u0926\u094b \u092f\u093e \u0926\u094b \u0938\u0947 \u0905\u0927\u093f\u0915 \u092a\u0942\u0930\u094d\u0935 \u092a\u0926 \u0938\u092e\u094d\u092e\u093f\u0932\u093f\u0924 \u0939\u094b \u0938\u0915\u0924\u0947 \u0939\u0948\u0902\u0964 \u0938\u092c\u0938\u0947 \u092a\u094d\u0930\u0938\u093f\u0926\u094d\u0927 \u0909\u0926\u093e\u0939\u0930\u0923 V1,V2,V3,... \u0939\u0948, \u091c\u0939\u093e\u0901 V1=1, V2=2 \u0914\u0930 Vn=Vn-1+Vn-2, \u091c\u092c n\u22653 \u0939\u094b\u0964<br>\nV3=3, V4=5, V5=8 \u2192 1,2,3,5,8,13,21,34,...<br>\n\u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u092a\u0926 \u092a\u0942\u0930\u094d\u0935 \u0915\u0947 \u0926\u094b \u092a\u0926\u094b\u0902 \u0915\u093e \u092f\u094b\u0917\u0964 \u0906\u0917\u093e\u092e\u0940 \u0926\u094b 55,89\u0964<br>\n\u092f\u0939 \u0935\u093f\u0930\u0939\u093e\u0902\u0915-\u092b\u093f\u092c\u094b\u0928\u093e\u091a\u0940 \u0905\u0928\u0941\u0915\u094d\u0930\u092e \u0939\u0948! \u0938\u0930\u094d\u0935\u092a\u094d\u0930\u0925\u092e \u0906\u091a\u093e\u0930\u094d\u092f \u0935\u093f\u0930\u0939\u093e\u0902\u0915 \u0926\u094d\u0935\u093e\u0930\u093e \u0938\u093e\u0924\u0935\u0940\u0902 \u0936\u0924\u093e\u092c\u094d\u0926\u0940 \u092e\u0947\u0902 \u0935\u0943\u0924\u094d\u0924\u091c\u093e\u0924\u093f \u0938\u092e\u0941\u091a\u094d\u091a\u092f \u092e\u0947\u0902 \u0932\u093f\u0916\u093e \u0917\u092f\u093e\u0964 \u0909\u0928\u094d\u0939\u094b\u0902\u0928\u0947 \u0907\u0938\u0915\u093e \u0905\u0928\u094d\u0935\u0947\u0937\u0923 \u092a\u094d\u0930\u093e\u0915\u0943\u0924 \u092e\u0940\u091f\u0930 \u0914\u0930 \u0915\u0935\u093f\u0924\u093e \u0915\u0947 \u0938\u0902\u0926\u0930\u094d\u092d \u092e\u0947\u0902 \u0915\u093f\u092f\u093e \u0925\u093e\u0964 \u0907\u0938\u0915\u093e \u0935\u093f\u0938\u094d\u0924\u0943\u0924 \u0905\u0927\u094d\u092f\u092f\u0928 \u092d\u093e\u0937\u093e\u0935\u093f\u0926-\u0917\u0923\u093f\u0924\u091c\u094d\u091e \u0917\u094b\u092a\u093e\u0932 (\u0932\u0917\u092d\u0917 1135) \u0914\u0930 \u0939\u0947\u092e\u091a\u0902\u0926\u094d\u0930 (\u0932\u0917\u092d\u0917 1150) \u0926\u094d\u0935\u093e\u0930\u093e \u0915\u093f\u092f\u093e \u0917\u092f\u093e\u0964 \u0906\u0917\u0947 \u091a\u0932\u0915\u0930 \u0907\u091f\u0932\u0940 \u0915\u0947 \u0917\u0923\u093f\u0924\u091c\u094d\u091e \u092b\u093f\u092c\u094b\u0928\u093e\u091a\u0940 (\u0932\u0917\u092d\u0917 1200) \u0926\u094d\u0935\u093e\u0930\u093e \u092d\u0940 \u0905\u0927\u094d\u092f\u092f\u0928 \u0915\u093f\u092f\u093e \u0917\u092f\u093e\u0964<br>\n\n<\/div><\/div>\n\n<div id=\"a8-5\" style=\"display:none;\"><div style=\"background:#ff6f00;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udfe0 \u092d\u093e\u0917 5\/5 - \u0938\u093e\u0930\u093e\u0902\u0936 \u0938\u0942\u0924\u094d\u0930 FINAL<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:14px;line-height:2;height:560px;overflow-y:auto;\">\n\n<div style=\"background:#e3f2fd;border:2px solid #1565c0;padding:10px;border-radius:10px;text-align:center;font-weight:bold;\">\n<b>\u0905\u0927\u094d\u092f\u093e\u092f 8.1-8.3 FINAL \u0936\u0941\u0926\u094d\u0927 \u0938\u093e\u0930\u093e\u0902\u0936<\/b><br><br>\n\u0905\u0928\u0941\u0915\u094d\u0930\u092e = \u0915\u094d\u0930\u092e\u093f\u0915 \u0938\u0942\u091a\u0940 t1,t2,t3...<br>\n\u092a\u0930\u093f\u092e\u093f\u0924 5 \u092a\u0926, \u0905\u092a\u0930\u093f\u092e\u093f\u0924 ...<br>\n\u092a\u094d\u0930\u093e\u0915\u0943\u0924 n, \u0935\u093f\u0937\u092e 2n-1, \u0924\u094d\u0930\u093f\u092d\u0941\u091c\u093e\u0915\u093e\u0930 n(n+1)\/2, \u0935\u0930\u094d\u0917 n\u00b2<br>\n\u0924\u094d\u0930\u093f\u092d\u0941\u091c\u093e\u0915\u093e\u0930 \u092f\u094b\u0917 1..n, \u0935\u0930\u094d\u0917 \u0935\u093f\u0937\u092e \u092f\u094b\u0917<br>\n\u0935\u094d\u092f\u093e\u092a\u0915 \u0938\u0942\u0924\u094d\u0930 tn=n \u0915\u093e \u092b\u0932\u0928 - \u0938\u0940\u0927\u0947 \u0915\u094b\u0908 \u092d\u0940 \u092a\u0926<br>\n\u0909\u0926\u093e \u0935\u093f\u0937\u092e tn=2n-1, sn=5n-2 \u092a\u094d\u0930\u0925\u092e 3,8,13...<br>\n\u092a\u0941\u0928\u0930\u093e\u0935\u0930\u094d\u0924\u0940 tn=tn-1+d - \u092a\u0942\u0930\u094d\u0935 \u092a\u0926 \u0938\u0947 \u0905\u0917\u0932\u093e<br>\n\u0909\u0926\u093e tn=tn-1+3, un=2un-1+3 \u21921,5,13,29<br>\nsn=sn-1(sn-1-1) \u21923,6,30,870<br>\n\u092b\u093f\u092c\u094b\u0928\u093e\u091a\u0940 Vn=Vn-1+Vn-2 1,2,3,5,8,13...<br>\n\u0935\u093f\u0930\u0939\u093e\u0902\u0915 7\u0935\u0940\u0902 \u0938\u0926\u0940 \u092d\u093e\u0930\u0924, \u0939\u0947\u092e\u091a\u0902\u0926\u094d\u0930, \u092b\u093f\u092c\u094b\u0928\u093e\u091a\u0940\n<\/div><br>\n\n<b>\u092a\u094d\u0930\u0936\u094d\u0928:<\/b> \u0935\u094d\u092f\u0902\u091c\u0915 tn=3n-7 (i) \u092a\u0939\u0932\u093e, \u0926\u0942\u0938\u0930\u093e, \u0924\u0940\u0938\u0930\u093e, 12\u0935\u093e\u0901, 18\u0935\u093e\u0901, 50\u0935\u093e\u0901 \u092a\u0926 \u091c\u094d\u091e\u093e\u0924 \u0915\u0940\u091c\u093f\u090f (ii) \u0915\u094c\u0928 \u0938\u093e \u092a\u0926 332 \u0939\u0948? (iii) \u0915\u094d\u092f\u093e 557 \u092a\u0926 \u0939\u0948?<br>\n\u0939\u0932: (i) t1=-4, t2=-1, t3=2, t12=29, t18=47, t50=143<br>\n(ii) 3n-7=332\u2192n=113 \u2192113\u0935\u093e\u0901 \u092a\u0926<br>\n(iii) 3n-7=557\u2192n=188 \u2192 \u0939\u093e\u0901 188\u0935\u093e\u0901 \u092a\u0926<br>\n\n<\/div><\/div>\n<\/div>\n\n<div style=\"background:#0a5c4b;padding:12px;display:flex;justify-content:space-between;align-items:center;\">\n<button type=\"button\" id=\"a8p\" style=\"background:#ffcc00;color:#000;border:3px solid #fff;border-radius:30px;padding:10px 22px;font-weight:bold;cursor:pointer;min-width:100px;\">&larr; \u092a\u0940\u091b\u0947<\/button>\n<span id=\"a8i\" style=\"background:#fff;color:#0a5c4b;padding:8px 20px;border-radius:20px;font-weight:bold;\">1 \/ 5<\/span>\n<button type=\"button\" id=\"a8n\" style=\"background:#00e676;color:#000;border:3px solid #fff;border-radius:30px;padding:10px 22px;font-weight:bold;cursor:pointer;min-width:100px;\">\u0906\u0917\u0947 &rarr;<\/button>\n<\/div>\n<\/div>\n\n<script>\n(function(){\nvar c=1,t=5;\nfunction s(n){\nfor(var i=1;i<=t;i++){\nvar el=document.getElementById('a8-'+i);\nif(el)el.style.display=(i===n?'block':'none');\n}\ndocument.getElementById('a8i').textContent=n+' \/ '+t;\nc=n;\n}\ndocument.getElementById('a8n').onclick=function(){var n=c+1;if(n>t)n=1;s(n);};\ndocument.getElementById('a8p').onclick=function(){var n=c-1;if(n<1)n=t;s(n);};\n})();\n<\/script><\/div><\/div><\/div><\/div><div id=\"pg-991-2\"  class=\"panel-grid panel-no-style\" ><div id=\"pgc-991-2-0\"  class=\"panel-grid-cell\" ><div id=\"panel-991-2-0-0\" class=\"widget_text so-panel widget widget_custom_html panel-first-child panel-last-child\" data-index=\"2\" ><div class=\"textwidget custom-html-widget\"><div id=\"Q81-ALL\" style=\"max-width:860px;margin:12px auto;border:4px solid #0f5e4b;border-radius:14px;background:#e6f4f1;font-family:'Noto Sans Devanagari',Arial,sans-serif;overflow:hidden;\">\n<div style=\"background:#0a5c4b;padding:10px;text-align:center;\"><span style=\"background:#fff;color:#0a3d33;padding:7px 20px;border-radius:20px;font-weight:bold;font-size:13px;\">\u092a\u094d\u0930\u0936\u094d\u0928\u093e\u0935\u0932\u0940 8.1 - Q1 \u0938\u0947 Q6 - \u092a\u0947\u091c 180<\/span><\/div>\n\n<div style=\"padding:10px 12px;\">\n<div id=\"q81-1\" style=\"display:block;\"><div style=\"background:#d32f2f;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udd34 \u092d\u093e\u0917 1\/4 - Q1 Q2 Q3<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:14px;line-height:1.9;height:560px;overflow-y:auto;\">\n\n<b>Q1. \u0909\u0938 \u0905\u0928\u0941\u0915\u094d\u0930\u092e \u0915\u0947 \u092a\u094d\u0930\u0925\u092e \u092a\u093e\u0901\u091a \u092a\u0926\u094b\u0902 \u0915\u094b \u091c\u094d\u091e\u093e\u0924 \u0915\u0940\u091c\u093f\u090f, \u091c\u093f\u0938\u0915\u093e n\u0935\u093e\u0901 \u092a\u0926 \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u092a\u094d\u0930\u0915\u093e\u0930 \u0938\u0947 \u0926\u093f\u092f\u093e \u0917\u092f\u093e \u0939\u0948\u2014 (i) tn=3n-4 (ii) tn=2-5n (iii) tn=n\u00b2-2n+3 \u091c\u0939\u093e\u0901 n\u22651 \u0939\u094b\u0964<\/b><br>\n\u0939\u0932:<br>\n(i) tn=3n-4<br>\nt1=-1, t2=2, t3=5, t4=8, t5=11 \u2192 -1,2,5,8,11<br><br>\n(ii) tn=2-5n<br>\nt1=-3, t2=-8, t3=-13, t4=-18, t5=-23 \u2192 -3,-8,-13,-18,-23<br><br>\n(iii) tn=n\u00b2-2n+3<br>\nt1=1-2+3=2, t2=4-4+3=3, t3=9-6+3=6, t4=16-8+3=11, t5=25-10+3=18<br>\n\u2192 2,3,6,11,18<br><br>\n\n<b>Q2. \u0905\u0928\u0941\u0915\u094d\u0930\u092e tn=5n-3 \u0939\u094b, \u091c\u092c\u0915\u093f n\u22651 \u0915\u0947 10\u0935\u0947\u0902 \u090f\u0935\u0902 15\u0935\u0947\u0902 \u092a\u0926\u094b\u0902 \u0915\u094b \u091c\u094d\u091e\u093e\u0924 \u0915\u0940\u091c\u093f\u090f\u0964<\/b><br>\nt10=50-3=47, t15=75-3=72<br>\n<span style=\"background:#e8f5e9;padding:3px 10px;border-radius:6px;font-weight:bold;\">\u0909\u0924\u094d\u0924\u0930 t10=47, t15=72<\/span><br><br>\n\n<b>Q3. \u0928\u093f\u0930\u094d\u0927\u093e\u0930\u093f\u0924 \u0915\u0940\u091c\u093f\u090f \u0915\u093f \u0915\u094d\u092f\u093e 97 \u0914\u0930 172 \u0905\u0928\u0941\u0915\u094d\u0930\u092e tn=5n-3 \u0915\u0947 \u092a\u0926 \u0939\u0948\u0902, \u091c\u092c\u0915\u093f n\u22651 \u0939\u0948\u0964<\/b><br>\n5n-3=97\u21925n=100\u2192n=20 \u2192 \u0939\u093e\u0901 20\u0935\u093e\u0901 \u092a\u0926<br>\n5n-3=172\u21925n=175\u2192n=35 \u2192 \u0939\u093e\u0901 35\u0935\u093e\u0901 \u092a\u0926<br>\n<span style=\"background:#e8f5e9;padding:3px 10px;border-radius:6px;font-weight:bold;\">\u0909\u0924\u094d\u0924\u0930 \u0926\u094b\u0928\u094b\u0902 \u092a\u0926 \u0939\u0948\u0902 20\u0935\u093e\u0901 \u0914\u0930 35\u0935\u093e\u0901<\/span><br>\n\n<\/div><\/div>\n\n<div id=\"q81-2\" style=\"display:none;\"><div style=\"background:#1565c0;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udd35 \u092d\u093e\u0917 2\/4 - Q4 Q5<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:14px;line-height:1.9;height:560px;overflow-y:auto;\">\n\n<b>Q4. \u0905\u0928\u0941\u0915\u094d\u0930\u092e tn=5n-3 \u091c\u0939\u093e\u0901 n\u22651 \u0939\u0948, \u092e\u0947\u0902 \u0915\u094c\u0928-\u0938\u093e \u092a\u0926 607 \u0939\u0948?<\/b><br>\n5n-3=607\u21925n=610\u2192n=122<br>\n<span style=\"background:#e8f5e9;padding:3px 10px;border-radius:6px;font-weight:bold;\">\u0909\u0924\u094d\u0924\u0930 122\u0935\u093e\u0901 \u092a\u0926<\/span><br><br>\n\n<b>Q5. \u090f\u0915 \u0905\u0928\u0941\u0915\u094d\u0930\u092e t1=-5, tn+1=tn+3 \u092a\u0941\u0928\u0930\u093e\u0935\u0930\u094d\u0924\u0940 \u0938\u0942\u0924\u094d\u0930 \u0926\u094d\u0935\u093e\u0930\u093e \u0926\u0930\u094d\u0936\u093e\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948, \u091c\u092c n\u22651 \u0939\u094b\u0964 \u0905\u0928\u0941\u0915\u094d\u0930\u092e \u0915\u0947 \u092a\u094d\u0930\u0925\u092e \u092a\u093e\u0901\u091a \u092a\u0926 \u091c\u094d\u091e\u093e\u0924 \u0915\u0940\u091c\u093f\u090f\u0964 \u0915\u094d\u092f\u093e \u0907\u0938 \u0905\u0928\u0941\u0915\u094d\u0930\u092e \u0915\u093e \u090f\u0915 \u092a\u0926 52 \u0939\u0948? \u092f\u0926\u093f \u0939\u093e\u0901, \u0924\u094b \u092f\u0939 \u0905\u0928\u0941\u0915\u094d\u0930\u092e \u0915\u093e \u0915\u094c\u0928-\u0938\u093e \u092a\u0926 \u0939\u0948?<\/b><br>\n\u0939\u0932:<br>\nt1=-5<br>\nt2=t1+3=-2<br>\nt3=t2+3=1<br>\nt4=t3+3=4<br>\nt5=t4+3=7<br>\n\u2192 -5,-2,1,4,7 AP d=3<br>\n\u0935\u094d\u092f\u093e\u092a\u0915 \u0938\u0942\u0924\u094d\u0930 tn = -5+(n-1)3 =3n-8<br>\n52 \u0939\u0948? 3n-8=52\u21923n=60\u2192n=20 \u2192 \u0939\u093e\u0901 20\u0935\u093e\u0901 \u092a\u0926<br>\n<span style=\"background:#e8f5e9;padding:3px 10px;border-radius:6px;font-weight:bold;\">\u0909\u0924\u094d\u0924\u0930 \u092a\u094d\u0930\u0925\u092e \u092a\u093e\u0901\u091a -5,-2,1,4,7, 52 20\u0935\u093e\u0901 \u092a\u0926<\/span><br>\n\n<\/div><\/div>\n\n<div id=\"q81-3\" style=\"display:none;\"><div style=\"background:#2e7d32;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udfe2 \u092d\u093e\u0917 3\/4 - Q6 \u091f\u094d\u0930\u093f\u092c\u094b\u0928\u093e\u091a\u0940<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:14px;line-height:1.9;height:560px;overflow-y:auto;\">\n\n<b>Q6. \u092e\u093e\u0928 \u0932\u0940\u091c\u093f\u090f, T1=1, T2=2, T3=4 \u0914\u0930 Tn=Tn-1+Tn-2+Tn-3 \u0939\u094b, \u091c\u092c\u0915\u093f n\u22654 \u0939\u0948\u0964 T4,T5,T6,T7 \u0914\u0930 T8 \u091c\u094d\u091e\u093e\u0924 \u0915\u0940\u091c\u093f\u090f\u0964<\/b><br>\n\u0939\u0932: \u091f\u094d\u0930\u093f\u092c\u094b\u0928\u093e\u091a\u0940 \u091c\u0948\u0938\u093e - \u0924\u0940\u0928 \u092a\u0926\u094b\u0902 \u0915\u093e \u092f\u094b\u0917<br>\nT1=1<br>\nT2=2<br>\nT3=4<br>\nT4=T3+T2+T1=4+2+1=7<br>\nT5=T4+T3+T2=7+4+2=13<br>\nT6=T5+T4+T3=13+7+4=24<br>\nT7=T6+T5+T4=24+13+7=44<br>\nT8=T7+T6+T5=44+24+13=81<br><br>\n\n<span style=\"background:#e8f5e9;padding:3px 10px;border-radius:6px;font-weight:bold;\">\u0909\u0924\u094d\u0924\u0930 T4=7, T5=13, T6=24, T7=44, T8=81<br>\u0905\u0928\u0941\u0915\u094d\u0930\u092e: 1,2,4,7,13,24,44,81,...<\/span><br><br>\n\n<div style=\"background:#fff9c4;border:2px solid #fbc02d;padding:8px;border-radius:8px;\">\n\u091f\u094d\u0930\u093f\u0915: \u092b\u093f\u092c\u094b\u0928\u093e\u091a\u0940 2 \u092a\u0926\u094b\u0902 \u0915\u093e \u092f\u094b\u0917, \u091f\u094d\u0930\u093f\u092c\u094b\u0928\u093e\u091a\u0940 3 \u092a\u0926\u094b\u0902 \u0915\u093e \u092f\u094b\u0917, \u091f\u0947\u091f\u094d\u0930\u093e\u092c\u094b\u0928\u093e\u091a\u0940 4 \u092a\u0926\u094b\u0902 \u0915\u093e\n<\/div>\n\n<\/div><\/div>\n\n<div id=\"q81-4\" style=\"display:none;\"><div style=\"background:#6a1b9a;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udfe3 \u092d\u093e\u0917 4\/4 - FINAL \u0938\u093e\u0930<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:14px;line-height:2;height:560px;overflow-y:auto;\">\n\n<div style=\"background:#fff9c4;border:3px solid #fbc02d;padding:10px;border-radius:10px;text-align:center;font-weight:bold;\">\n\u2705 \u092a\u094d\u0930\u0936\u094d\u0928\u093e\u0935\u0932\u0940 8.1 Q1-Q6 \u0936\u0941\u0926\u094d\u0927 \u0916\u0924\u094d\u092e<br><br>\nQ1 (i) 3n-4 \u2192 -1,2,5,8,11<br>\n(ii) 2-5n \u2192 -3,-8,-13,-18,-23<br>\n(iii) n\u00b2-2n+3 \u21922,3,6,11,18<br>\nQ2 tn=5n-3 t10=47 t15=72<br>\nQ3 97 20\u0935\u093e\u0901, 172 35\u0935\u093e\u0901 \u0926\u094b\u0928\u094b\u0902 \u092a\u0926 \u0939\u0948\u0902<br>\nQ4 607 122\u0935\u093e\u0901 \u092a\u0926<br>\nQ5 t1=-5 tn+1=tn+3 \u2192-5,-2,1,4,7, 52 20\u0935\u093e\u0901 \u092a\u0926<br>\nQ6 T1=1 T2=2 T3=4 Tn=Tn-1+Tn-2+Tn-3<br>\nT4=7 T5=13 T6=24 T7=44 T8=81\n<\/div>\n\n<\/div><\/div>\n<\/div>\n\n<div style=\"background:#0a5c4b;padding:12px;display:flex;justify-content:space-between;align-items:center;\">\n<button type=\"button\" id=\"q81p\" style=\"background:#ffcc00;color:#000;border:3px solid #fff;border-radius:30px;padding:10px 22px;font-weight:bold;cursor:pointer;min-width:100px;\">&larr; \u092a\u0940\u091b\u0947<\/button>\n<span id=\"q81i\" style=\"background:#fff;color:#0a5c4b;padding:8px 20px;border-radius:20px;font-weight:bold;\">1 \/ 4<\/span>\n<button type=\"button\" id=\"q81n\" style=\"background:#00e676;color:#000;border:3px solid #fff;border-radius:30px;padding:10px 22px;font-weight:bold;cursor:pointer;min-width:100px;\">\u0906\u0917\u0947 &rarr;<\/button>\n<\/div>\n<\/div>\n\n<script>\n(function(){\nvar c=1,t=4;\nfunction s(n){\nfor(var i=1;i<=t;i++){\nvar el=document.getElementById('q81-'+i);\nif(el)el.style.display=(i===n?'block':'none');\n}\ndocument.getElementById('q81i').textContent=n+' \/ '+t;\nc=n;\n}\ndocument.getElementById('q81n').onclick=function(){var n=c+1;if(n>t)n=1;s(n);};\ndocument.getElementById('q81p').onclick=function(){var n=c-1;if(n<1)n=t;s(n);};\n})();\n<\/script><\/div><\/div><\/div><\/div><div id=\"pg-991-3\"  class=\"panel-grid panel-no-style\" ><div id=\"pgc-991-3-0\"  class=\"panel-grid-cell\" ><div id=\"panel-991-3-0-0\" class=\"widget_text so-panel widget widget_custom_html panel-first-child panel-last-child\" data-index=\"3\" ><div class=\"textwidget custom-html-widget\"><div id=\"CH84-85\" style=\"max-width:860px;margin:12px auto;border:4px solid #0f5e4b;border-radius:14px;background:#e6f4f1;font-family:'Noto Sans Devanagari',Arial,sans-serif;overflow:hidden;\">\n<div style=\"background:#0a5c4b;padding:10px;text-align:center;\"><span style=\"background:#fff;color:#0a3d33;padding:7px 20px;border-radius:20px;font-weight:bold;font-size:13px;\">8.4 AP \u0914\u0930 8.5 n \u092a\u094d\u0930\u093e\u0915\u0943\u0924 \u092f\u094b\u0917 - \u092a\u0947\u091c 181-185 \u0936\u0941\u0926\u094d\u0927<\/span><\/div>\n\n<div style=\"padding:10px 12px;\">\n<div id=\"s84-1\" style=\"display:block;\"><div style=\"background:#d32f2f;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udd34 \u092d\u093e\u0917 1\/5 - 8.4 \u0938\u092e\u093e\u0928\u094d\u0924\u0930 \u0936\u094d\u0930\u0947\u0922\u093c\u0940 \u092a\u0930\u093f\u091a\u092f<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:14px;line-height:2;height:560px;overflow-y:auto;\">\n\n<div style=\"background:#ffebee;border-left:4px solid #d32f2f;padding:8px 10px;font-weight:bold;\">8.4 \u0938\u092e\u093e\u0928\u094d\u0924\u0930 \u0936\u094d\u0930\u0947\u0922\u093c\u093f\u092f\u093e\u0901 \u0936\u0941\u0926\u094d\u0927<\/div>\n\u0905\u092c \u0924\u0915 \u0939\u092e\u0928\u0947 \u0905\u0927\u094d\u092f\u092f\u0928 \u0915\u093f\u092f\u093e \u0915\u093f \u0905\u0928\u0941\u0915\u094d\u0930\u092e, \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0940 \u0910\u0938\u0940 \u0915\u094d\u0930\u092e\u093f\u0915 \u0938\u0942\u091a\u0940 \u0939\u0948, \u091c\u094b \u0915\u093f\u0938\u0940 \u0935\u093f\u0936\u0947\u0937 \u0938\u0942\u0924\u094d\u0930 \u0915\u093e \u092a\u093e\u0932\u0928 \u0915\u0930\u0924\u0940 \u0939\u0948\u0964 \u092f\u0926\u094d\u092f\u092a\u093f, \u0910\u0938\u0947 \u0905\u0928\u0941\u0915\u094d\u0930\u092e \u092d\u0940 \u0939\u094b \u0938\u0915\u0924\u0947 \u0939\u0948\u0902, \u091c\u0948\u0938\u0947 \u0915\u093f \u0905\u092d\u093e\u091c\u094d\u092f \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u093e \u0905\u0928\u0941\u0915\u094d\u0930\u092e, \u091c\u093f\u0938\u0915\u0947 \u0938\u0942\u0924\u094d\u0930 \u092e\u0947\u0902 \u0915\u094b\u0908 \u0928\u093f\u092f\u092e\u093f\u0924\u0924\u093e \u0928\u0939\u0940\u0902 \u0939\u094b\u0924\u0940\u0964<br>\n\u0939\u092e \u0907\u0938 \u0905\u0928\u0941\u092d\u093e\u0917 \u092e\u0947\u0902, \u090f\u0915 \u0935\u093f\u0936\u0947\u0937 \u092a\u094d\u0930\u0915\u093e\u0930 \u0915\u0947 \u0905\u0928\u0941\u0915\u094d\u0930\u092e\u094b\u0902 \u0915\u093e \u0905\u0928\u094d\u0935\u0947\u0937\u0923 \u0915\u0930\u0947\u0902\u0917\u0947, \u091c\u094b \u0938\u092e\u093e\u0928\u094d\u0924\u0930 \u0936\u094d\u0930\u0947\u0922\u093c\u0940 \u0915\u0939\u0932\u093e\u0924\u0947 \u0939\u0948\u0902\u0964<br><br>\n\n<b>\u091a\u093f\u0924\u094d\u0930 8.3 \u0935\u0930\u094d\u0917\u094b\u0902 \u0915\u093e \u092c\u0922\u093c\u0924\u093e \u0939\u0941\u0906 \u092a\u094d\u0930\u0924\u093f\u0930\u0942\u092a:<\/b><br>\n\u091a\u0930\u0923 1 2 3 4 \u092e\u0947\u0902 \u091b\u094b\u091f\u0947 \u0935\u0930\u094d\u0917\u094b\u0902 \u0915\u0940 \u0938\u0902\u0916\u094d\u092f\u093e 1,5,9,13 \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0939\u094b\u0924\u0940 \u0939\u0948\u0964<br>\n\u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u091a\u0930\u0923 \u092e\u0947\u0902 \u092a\u093f\u091b\u0932\u0947 \u091a\u0930\u0923 \u0915\u0947 \u092a\u094d\u0930\u0924\u093f\u0930\u0942\u092a \u0915\u0947 \u0915\u094b\u0928\u0947 \u0915\u0947 \u092c\u093f\u0902\u0926\u0941\u0913\u0902 \u0938\u0947 4 \u0935\u0930\u094d\u0917 \u091c\u0941\u0921\u093c \u091c\u093e\u0924\u0947 \u0939\u0948\u0902\u0964<br>\n\u0906\u0917\u093e\u092e\u0940 \u091a\u0930\u0923\u094b\u0902 \u092e\u0947\u0902 \u0935\u0930\u094d\u0917\u094b\u0902 \u0915\u0940 \u0938\u0902\u0916\u094d\u092f\u093e \u0915\u094b \u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930 \u0932\u093f\u0916\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948\u2014<br>\n1, 1+4, 1+4+4, 1+4+4+4,...<br>\n\u0907\u0938\u0947 \u092a\u0941\u0928: 1, 1+1\u00d74, 1+2\u00d74, 1+3\u00d74,...<br>\nn\u0935\u093e\u0901 \u092a\u0926 tn = 1+(n-1)\u00d74 =4n-3<br>\n\u092a\u094d\u0930\u0925\u092e \u091b\u0939 \u092a\u0926 1,5,9,13,17,21 \u0905\u0902\u0924\u0930 \u0905\u091a\u0930 4<br><br>\n\n<div style=\"background:#fff9c4;border:2px solid #fbc02d;padding:8px;border-radius:8px;text-align:center;font-weight:bold;\">\n\u092a\u0930\u093f\u092d\u093e\u0937\u093e \u0936\u0941\u0926\u094d\u0927: \u0910\u0938\u0947 \u0905\u0928\u0941\u0915\u094d\u0930\u092e \u091c\u093f\u0928\u092e\u0947\u0902 \u0926\u094b \u0915\u094d\u0930\u092e\u093f\u0915 \u092a\u0926\u094b\u0902 \u0915\u0947 \u092c\u0940\u091a \u0905\u0902\u0924\u0930 \u0905\u091a\u0930 \u0939\u094b, \u0938\u092e\u093e\u0928\u094d\u0924\u0930 \u0936\u094d\u0930\u0947\u0922\u093c\u0940 (AP) \u0915\u0939\u0932\u093e\u0924\u0947 \u0939\u0948\u0902\u0964\n<\/div>\n\n<\/div><\/div>\n\n<div id=\"s84-2\" style=\"display:none;\"><div style=\"background:#1565c0;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udd35 \u092d\u093e\u0917 2\/5 - AP \u0938\u0942\u0924\u094d\u0930<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:14px;line-height:1.9;height:560px;overflow-y:auto;\">\n\n\u0927\u094d\u092f\u093e\u0928 \u0926\u0940\u091c\u093f\u090f \u0915\u093f \u090a\u092a\u0930 \u0926\u093f\u090f \u0917\u090f \u0905\u0928\u0941\u0915\u094d\u0930\u092e \u0915\u0947 n\u0935\u0947\u0902 \u092a\u0926 tn=1+(n-1)\u00d74 \u092e\u0947\u0902 1 \u092a\u094d\u0930\u0925\u092e \u092a\u0926 \u0939\u0948 \u0914\u0930 4 \u0938\u093e\u0930\u094d\u0935 \u0905\u0902\u0924\u0930 \u0939\u0948\u0964 \u0907\u0938\u0940 \u092a\u094d\u0930\u0915\u093e\u0930, \u0905\u0928\u0941\u0915\u094d\u0930\u092e 1,4,7,10,... \u0915\u093e n\u0935\u093e\u0901 \u092a\u0926 tn=1+(n-1)\u00d73 \u0939\u0948, \u091c\u093f\u0938\u092e\u0947\u0902 1 \u092a\u094d\u0930\u0925\u092e \u092a\u0926 \u0914\u0930 3 \u0938\u093e\u0930\u094d\u0935 \u0905\u0902\u0924\u0930\u0964 \u0905\u0928\u0941\u0915\u094d\u0930\u092e 11,7,3,-1,-5,... \u092e\u0947\u0902 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u092e\u0947\u0902 4 \u0915\u0940 \u0915\u092e\u0940 \u0939\u094b\u0924\u0940 \u091c\u093e\u0924\u0940 \u0939\u0948\u0964 \u092f\u0939 \u092d\u0940 AP \u0939\u0948, \u092a\u094d\u0930\u0925\u092e \u092a\u0926 11 \u0914\u0930 \u0938\u093e\u0930\u094d\u0935 \u0905\u0902\u0924\u0930 -4\u0964<br><br>\n\n<div style=\"background:#e3f2fd;border:3px solid #1565c0;padding:10px;border-radius:10px;text-align:center;font-weight:bold;font-size:15px;\">\n\u0938\u093e\u092e\u093e\u0928\u094d\u092f\u0924\u0903 AP \u0915\u0940 \u0935\u094d\u092f\u093e\u0916\u094d\u092f\u093e:<br>\na, a+d, a+2d, a+3d,..., a+(n-1)\u00d7d<br>\n\u091c\u0939\u093e\u0901 a \u092a\u094d\u0930\u0925\u092e \u092a\u0926, d \u0938\u093e\u0930\u094d\u0935 \u0905\u0902\u0924\u0930<br>\nn\u0935\u093e\u0901 \u092a\u0926 tn = a+(n-1)d\n<\/div><br>\n\n<b>\u092a\u094d\u0930\u0936\u094d\u0928:<\/b> \u0938\u0924\u094d\u092f\u093e\u092a\u093f\u0924 \u0915\u0940\u091c\u093f\u090f \u0915\u093f \u0928\u0940\u091a\u0947 \u0926\u093f\u090f \u0917\u090f \u0905\u0928\u0941\u0915\u094d\u0930\u092e \u0938\u092e\u093e\u0928\u094d\u0924\u0930 \u0936\u094d\u0930\u0947\u0922\u093c\u093f\u092f\u093e\u0901 \u0939\u0948\u0902 \u0914\u0930 \u0909\u0928\u0915\u0947 n\u0935\u0947\u0902 \u092a\u0926 \u0932\u093f\u0916\u093f\u090f\u0964 \u0915\u094d\u0930\u092e\u093f\u0924 \u092f\u0941\u0917\u094d\u092e\u094b\u0902 \u0915\u094b \u0906\u0932\u0947\u0916 \u092a\u0930 \u092a\u094d\u0930\u0926\u0930\u094d\u0936\u093f\u0924 \u0915\u0930\u0928\u0947 \u092a\u0930 \u0906\u092a \u0915\u094d\u092f\u093e \u0905\u0935\u0932\u094b\u0915\u0928 \u0915\u0930\u0924\u0947 \u0939\u0948\u0902?<br>\n(i) 2,5,8,11,... (ii) -5,-1,3,7,...<br>\n\u0939\u0932: (i) d=3 AP tn=2+(n-1)3=3n-1, \u092c\u093f\u0902\u0926\u0941 (1,2)(2,5)(3,8) \u0938\u0930\u0932 \u0930\u0947\u0916\u093e \u092a\u0930<br>\n(ii) d=4 AP tn=-5+(n-1)4=4n-9, \u092c\u093f\u0902\u0926\u0941 \u0938\u0930\u0932 \u0930\u0947\u0916\u093e \u092a\u0930<br>\n\u0928\u093f\u0937\u094d\u0915\u0930\u094d\u0937: AP \u0915\u0947 \u092c\u093f\u0902\u0926\u0941 \u0939\u092e\u0947\u0936\u093e \u0938\u0930\u0932 \u0930\u0947\u0916\u093e \u092a\u0930 \u0939\u094b\u0924\u0947 \u0939\u0948\u0902 - \u0930\u0948\u0916\u093f\u0915 \u092a\u094d\u0930\u0924\u093f\u0930\u0942\u092a<br><br>\n\n<b>8.4.1 AP \u0915\u093e \u0926\u0943\u0936\u094d\u092f\u093e\u0902\u0915\u0928 \u0936\u0941\u0926\u094d\u0927:<\/b><br>\n\u091a\u0930\u0923 \u0938\u0902\u0916\u094d\u092f\u093e 1 2 3 4 5 ... n<br>\n\u0935\u0930\u094d\u0917\u094b\u0902 \u0915\u0940 \u0938\u0902\u0916\u094d\u092f\u093e 1 5 9 13 17 ... 4n-3<br>\n\u0915\u094d\u0930\u092e\u093f\u0924 \u092f\u0941\u0917\u094d\u092e (1,1)(2,5)(3,9)(4,13)(5,17) \u0906\u0932\u0947\u0916 \u092a\u0930 \u0938\u0930\u0932 \u0930\u0947\u0916\u093e \u092a\u0930 - \u091a\u093f\u0924\u094d\u0930 8.4<br>\n\n<\/div><\/div>\n\n<div id=\"s84-3\" style=\"display:none;\"><div style=\"background:#2e7d32;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udfe2 \u092d\u093e\u0917 3\/5 - AP \u092a\u094d\u0930\u0936\u094d\u0928 \u0914\u0930 \u0909\u0926\u093e\u0939\u0930\u0923 5<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:14px;line-height:1.9;height:560px;overflow-y:auto;\">\n\n<b>\u092a\u094d\u0930\u0936\u094d\u0928:<\/b> \u0938\u0942\u0924\u094d\u0930 tn=a+(n-1)d \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0915\u0930\u0924\u0947 \u0939\u0941\u090f \u0928\u0940\u091a\u0947 \u0926\u0940 \u0917\u0908 \u0938\u092e\u093e\u0928\u094d\u0924\u0930 \u0936\u094d\u0930\u0947\u0922\u093c\u093f\u092f\u094b\u0902 \u0915\u0947 n\u0935\u0947\u0902 \u092a\u0926 \u091c\u094d\u091e\u093e\u0924 \u0915\u0940\u091c\u093f\u090f\u0964<br>\n(i) 1\/2,5\/2,9\/2,13\/2,... (ii) 1.5,3.5,5.5,7.5,...<br>\n\u0939\u0932: (i) a=1\/2 d=2 tn=1\/2+(n-1)2=2n-3\/2<br>\n(ii) a=1.5 d=2 tn=1.5+(n-1)2=2n-0.5<br><br>\n\n\u0927\u094d\u092f\u093e\u0928 \u0926\u0940\u091c\u093f\u090f tn=a+(n-1)d \u0935\u094d\u092f\u093e\u092a\u0915 \u0938\u0942\u0924\u094d\u0930 \u0939\u0948\u0964 \u092a\u0941\u0928\u0930\u093e\u0935\u0930\u094d\u0924\u0940 \u0938\u0942\u0924\u094d\u0930 t1=a, tn=tn-1+d \u091c\u092c n\u22652<br>\nAP 1,5,9,13,17,... \u0915\u094b t1=1, tn=tn-1+4<br><br>\n\n<b>\u0909\u0926\u093e\u0939\u0930\u0923 5 \u0936\u0941\u0926\u094d\u0927:<\/b> \u090f\u0915 \u0935\u094d\u092f\u0915\u094d\u0924\u093f \u0936\u0939\u0930 \u092e\u0947\u0902 \u0918\u0942\u092e\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u091f\u0948\u0915\u094d\u0938\u0940 \u0915\u093f\u0930\u093e\u090f \u092a\u0930 \u0932\u0947\u0924\u093e \u0939\u0948\u0964 \u091f\u0948\u0915\u094d\u0938\u0940 \u0915\u0902\u092a\u0928\u0940 \u090f\u0915 \u0928\u093f\u0936\u094d\u091a\u093f\u0924 \u0906\u0930\u0915\u094d\u0937\u093f\u0924 \u0936\u0941\u0932\u094d\u0915 \u20b9200 \u0932\u0947\u0924\u0940 \u0939\u0948 \u0914\u0930 \u20b940 \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u0915\u093f\u092e\u0940 \u092f\u093e\u0924\u094d\u0930\u093e \u0915\u0947 \u0932\u093f\u090f \u0932\u0947\u0924\u0940 \u0939\u0948\u0964 1 \u0915\u093f\u092e\u0940,2 \u0915\u093f\u092e\u0940,3 \u0915\u093f\u092e\u0940 \u0906\u0926\u093f \u092f\u093e\u0924\u094d\u0930\u093e \u0915\u0947 \u0932\u093f\u090f \u0915\u0941\u0932 \u0915\u093f\u0930\u093e\u090f \u0915\u094b \u0926\u0930\u094d\u0936\u093e\u0928\u0947 \u0935\u093e\u0932\u093e \u0905\u0928\u0941\u0915\u094d\u0930\u092e \u0932\u093f\u0916\u093f\u090f\u0964 \u092f\u0926\u093f \u0935\u0939 \u0935\u094d\u092f\u0915\u094d\u0924\u093f 10 \u0915\u093f\u092e\u0940 \u0932\u0902\u092c\u0940 \u092f\u093e\u0924\u094d\u0930\u093e \u0915\u0930\u0924\u093e \u0939\u0948 \u0924\u094b \u0915\u0941\u0932 \u0915\u093f\u0930\u093e\u092f\u093e \u0915\u093f\u0924\u0928\u093e \u0939\u094b\u0917\u093e?<br>\n1 \u0915\u093f\u092e\u0940 \u092c\u093e\u0926 \u20b9200+\u20b940=\u20b9240, 2 \u0915\u093f\u092e\u0940 \u20b9280, 3 \u0915\u093f\u092e\u0940 \u20b9320... AP 240,280,320... a=240 d=40<br>\nn\u0935\u093e\u0901 \u092a\u0926 240+(n-1)40=200+40n, \u091c\u0939\u093e\u0901 n \u0915\u093f\u092e\u0940 \u092e\u0947\u0902 \u0924\u092f \u092f\u093e\u0924\u094d\u0930\u093e<br>\n10 \u0915\u093f\u092e\u0940: 200+400=\u20b9600<br>\n<span style=\"background:#e8f5e9;padding:3px 10px;border-radius:6px;font-weight:bold;\">\u0909\u0924\u094d\u0924\u0930 \u20b9600<\/span><br>\n\n<\/div><\/div>\n\n<div id=\"s84-4\" style=\"display:none;\"><div style=\"background:#6a1b9a;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udfe3 \u092d\u093e\u0917 4\/5 - 8.5 \u092a\u094d\u0930\u0925\u092e n \u092a\u094d\u0930\u093e\u0915\u0943\u0924 \u092f\u094b\u0917<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:14px;line-height:1.9;height:560px;overflow-y:auto;\">\n\n<div style=\"background:#f3e5f5;border-left:4px solid #6a1b9a;padding:8px 10px;font-weight:bold;\">8.5 \u092a\u094d\u0930\u0925\u092e n \u092a\u094d\u0930\u093e\u0915\u0943\u0924 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u093e \u092f\u094b\u0917 \u0936\u0941\u0926\u094d\u0927<\/div>\n\u0907\u0938 \u0905\u0928\u0941\u092d\u093e\u0917 \u092e\u0947\u0902, \u0939\u092e \u092a\u094d\u0930\u0925\u092e n \u092a\u094d\u0930\u093e\u0915\u0943\u0924 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0947 \u092f\u094b\u0917 \u0915\u0947 \u0932\u093f\u090f \u090f\u0915 \u0938\u0930\u0932 \u0915\u093f\u0902\u0924\u0941 \u092e\u0939\u0924\u094d\u0935\u092a\u0942\u0930\u094d\u0923 \u0938\u0942\u0924\u094d\u0930 \u0915\u093e \u0928\u093f\u0917\u092e\u0928 \u0915\u0930\u0947\u0902\u0917\u0947\u0964<br><br>\n\n<b>\u0935\u093f\u0927\u093f \u0936\u0941\u0926\u094d\u0927:<\/b> \u092e\u093e\u0928 \u0932\u0940\u091c\u093f\u090f S \u092a\u094d\u0930\u0925\u092e \u0926\u0938 \u092a\u094d\u0930\u093e\u0915\u0943\u0924 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0947 \u092f\u094b\u0917 \u0915\u094b \u0926\u0930\u094d\u0936\u093e\u0924\u093e \u0939\u0948\u0964<br>\nS=1+2+3+4+5+6+7+8+9+10<br>\nS=10+9+8+7+6+5+4+3+2+1 (\u0935\u094d\u092f\u0941\u0924\u094d\u0915\u094d\u0930\u092e)<br>\n\u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u092f\u0941\u0917\u094d\u092e \u092f\u094b\u0917 11 \u2192 2S=11\u00d710=110 \u2192 S=55<br><br>\n\n\u0907\u0938\u0940 \u0935\u093f\u0927\u093f \u0926\u094d\u0935\u093e\u0930\u093e 1+2+3+...+100 \u2192 2S=101\u00d7100 \u2192 S=5050<br><br>\n\n<div style=\"background:#fff9c4;border:3px solid #fbc02d;padding:10px;border-radius:10px;text-align:center;font-weight:bold;font-size:16px;\">\n\u0938\u093e\u092e\u093e\u0928\u094d\u092f \u0938\u0942\u0924\u094d\u0930:<br>\nS=1+2+...+n<br>\nS=n+(n-1)+...+1<br>\n2S=n(n+1)<br>\nS=n(n+1)\/2<br>\nSn=n(n+1)\/2\n<\/div><br>\n\n\u091a\u093f\u0924\u094d\u0930 8.5 \u092e\u0947\u0902 1+2+3+4+5+6 \u092f\u094b\u0917: 2\u00d7(\u092f\u094b\u0917)=7\u00d76 \u2192 \u0906\u092f\u0924\u093e\u0915\u093e\u0930 \u0935\u094d\u092f\u0935\u0938\u094d\u0925\u093e<br>\n2\u00d7(1..10)=10\u00d711<br><br>\n\n\u092f\u0939 \u092a\u0930\u093f\u0923\u093e\u092e \u092a\u094d\u0930\u0925\u092e \u091c\u094d\u091e\u093e\u0924 \u0932\u093f\u0916\u093f\u0924 \u0909\u0932\u094d\u0932\u0947\u0916 \u0906\u0930\u094d\u092f\u092d\u091f\u094d\u091f \u0915\u0947 \u0917\u094d\u0930\u0902\u0925 \u0906\u0930\u094d\u092f\u092d\u091f\u0940\u092f \u0915\u0947 \u0905\u0927\u094d\u092f\u093e\u092f 2 \u0915\u0947 \u091b\u0902\u0926 19 \u092e\u0947\u0902 \u0926\u0947\u0916\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948 - \u092f\u094b\u0917 \u0915\u0940 \u0917\u0923\u0928\u093e \u092a\u0939\u0932\u0947 \u0914\u0930 \u0905\u0902\u0924\u093f\u092e \u092a\u0926\u094b\u0902 \u0915\u0947 \u092f\u094b\u0917 \u0915\u094b \u0926\u094b \u0938\u0947 \u0935\u093f\u092d\u093e\u091c\u093f\u0924 \u0915\u0930\u0915\u0947 (\u0914\u0938\u0924) \u0914\u0930 \u092a\u0926\u094b\u0902 \u0915\u0940 \u0938\u0902\u0916\u094d\u092f\u093e \u0938\u0947 \u0917\u0941\u0923\u093e \u0915\u0930\u0915\u0947 \u0915\u0940 \u091c\u093e \u0938\u0915\u0924\u0940 \u0939\u0948\u0964<br>\n\n<\/div><\/div>\n\n<div id=\"s84-5\" style=\"display:none;\"><div style=\"background:#ff6f00;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udfe0 \u092d\u093e\u0917 5\/5 - \u092f\u094b\u0917 \u0915\u093e \u0909\u092a\u092f\u094b\u0917<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:14px;line-height:2;height:560px;overflow-y:auto;\">\n\n<b>\u0909\u092a\u092f\u094b\u0917 \u0936\u0941\u0926\u094d\u0927:<\/b><br>\n\u0915\u094d\u092f\u093e \u0906\u092a \u0938\u0942\u0924\u094d\u0930 \u0915\u093e \u0909\u092a\u092f\u094b\u0917 S20, S50, S1000 \u091c\u094d\u091e\u093e\u0924 \u0915\u0930\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u0915\u0930 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902?<br>\nS20=20\u00d721\/2=210<br>\nS50=50\u00d751\/2=1275<br>\nS1000=1000\u00d71001\/2=500500<br><br>\n\n\u0915\u094d\u0930\u092e\u093e\u0917\u0924 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u093e \u092f\u094b\u0917 \u091c\u0948\u0938\u0947 25+26+...+58:<br>\n= (1+...+58)-(1+...+24) =S58-S24<br>\n=58\u00d759\/2 -24\u00d725\/2 =29\u00d759-12\u00d725=1711-300=1411<br><br>\n\n<b>\u0924\u094d\u0930\u093f\u092d\u0941\u091c\u093e\u0915\u093e\u0930 \u0938\u0902\u0916\u094d\u092f\u093e\u090f\u0901:<\/b> 1,3,6,10,15,... \u0905\u0928\u0941\u0915\u094d\u0930\u092e tn \u092a\u0930 \u092a\u0941\u0928: \u0935\u093f\u091a\u093e\u0930\u0964 \u0907\u0938\u0915\u093e n\u0935\u093e\u0901 \u092a\u0926 \u092a\u094d\u0930\u0925\u092e n \u092a\u094d\u0930\u093e\u0915\u0943\u0924 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u093e \u092f\u094b\u0917 \u0939\u0948\u0964<br>\ntn=n(n+1)\/2<br>\n10\u0935\u0940\u0902=55, 17\u0935\u0940\u0902=153, 80\u0935\u0940\u0902=3240<br><br>\n\n<div style=\"background:#fff9c4;border:3px solid #fbc02d;padding:10px;border-radius:10px;text-align:center;font-weight:bold;\">\n\u2705 8.4-8.5 FINAL<br>\nAP \u092a\u0930\u093f\u092d\u093e\u0937\u093e \u0905\u0902\u0924\u0930 \u0905\u091a\u0930 d<br>\ntn=a+(n-1)d<br>\n\u092a\u0941\u0928\u0930\u093e\u0935\u0930\u094d\u0924\u0940 t1=a, tn=tn-1+d<br>\n\u0909\u0926\u093e\u0939\u0930\u0923 \u091f\u0948\u0915\u094d\u0938\u0940 200+40n 10 \u0915\u093f\u092e\u0940 \u20b9600<br>\n\u091a\u093f\u0924\u094d\u0930 \u0935\u0930\u094d\u0917 1,5,9,13 AP d4 tn=4n-3 \u0938\u0930\u0932 \u0930\u0947\u0916\u093e<br>\n\u092f\u094b\u0917 Sn=n(n+1)\/2 \u0906\u0930\u094d\u092f\u092d\u091f\u094d\u091f<br>\nS10=55 S100=5050<br>\n25..58 \u092f\u094b\u0917 S58-S24=1411<br>\n\u0924\u094d\u0930\u093f\u092d\u0941\u091c\u093e\u0915\u093e\u0930 tn=n(n+1)\/2\n<\/div>\n\n<\/div><\/div>\n<\/div>\n\n<div style=\"background:#0a5c4b;padding:12px;display:flex;justify-content:space-between;align-items:center;\">\n<button type=\"button\" id=\"s84p\" style=\"background:#ffcc00;color:#000;border:3px solid #fff;border-radius:30px;padding:10px 22px;font-weight:bold;cursor:pointer;min-width:100px;\">&larr; \u092a\u0940\u091b\u0947<\/button>\n<span id=\"s84i\" style=\"background:#fff;color:#0a5c4b;padding:8px 20px;border-radius:20px;font-weight:bold;\">1 \/ 5<\/span>\n<button type=\"button\" id=\"s84n\" style=\"background:#00e676;color:#000;border:3px solid #fff;border-radius:30px;padding:10px 22px;font-weight:bold;cursor:pointer;min-width:100px;\">\u0906\u0917\u0947 &rarr;<\/button>\n<\/div>\n<\/div>\n\n<script>\n(function(){\nvar c=1,t=5;\nfunction s(n){\nfor(var i=1;i<=t;i++){\nvar el=document.getElementById('s84-'+i);\nif(el)el.style.display=(i===n?'block':'none');\n}\ndocument.getElementById('s84i').textContent=n+' \/ '+t;\nc=n;\n}\ndocument.getElementById('s84n').onclick=function(){var n=c+1;if(n>t)n=1;s(n);};\ndocument.getElementById('s84p').onclick=function(){var n=c-1;if(n<1)n=t;s(n);};\n})();\n<\/script><\/div><\/div><\/div><\/div><div id=\"pg-991-4\"  class=\"panel-grid panel-no-style\" ><div id=\"pgc-991-4-0\"  class=\"panel-grid-cell\" ><div id=\"panel-991-4-0-0\" class=\"widget_text so-panel widget widget_custom_html panel-first-child panel-last-child\" data-index=\"4\" ><div class=\"textwidget custom-html-widget\"><div id=\"Q82-ALL\" style=\"max-width:860px;margin:12px auto;border:4px solid #0f5e4b;border-radius:14px;background:#e6f4f1;font-family:'Noto Sans Devanagari',Arial,sans-serif;overflow:hidden;\">\n<div style=\"background:#0a5c4b;padding:10px;text-align:center;\"><span style=\"background:#fff;color:#0a3d33;padding:7px 20px;border-radius:20px;font-weight:bold;font-size:13px;\">\u092a\u094d\u0930\u0936\u094d\u0928\u093e\u0935\u0932\u0940 8.2 - Q1-Q7 - \u092a\u0947\u091c 186<\/span><\/div>\n\n<div style=\"padding:10px 12px;\">\n<div id=\"q82-1\" style=\"display:block;\"><div style=\"background:#d32f2f;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udd34 \u092d\u093e\u0917 1\/4 - Q1 Q2 Q3<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:14px;line-height:1.9;height:560px;overflow-y:auto;\">\n\n<b>Q1. \u0938\u092e\u093e\u0928\u094d\u0924\u0930 \u0936\u094d\u0930\u0947\u0922\u093c\u0940 3,8,13,18,... \u0915\u0947 10\u0935\u0947\u0902 \u0914\u0930 26\u0935\u0947\u0902 \u092a\u0926 \u091c\u094d\u091e\u093e\u0924 \u0915\u0940\u091c\u093f\u090f\u0964<\/b><br>\na=3 d=5 tn=a+(n-1)d<br>\nt10=3+9\u00d75=3+45=48<br>\nt26=3+25\u00d75=3+125=128<br>\n<span style=\"background:#e8f5e9;padding:3px 10px;border-radius:6px;font-weight:bold;\">\u0909\u0924\u094d\u0924\u0930 48 \u0914\u0930 128<\/span><br><br>\n\n<b>Q2. \u0938\u092e\u093e\u0928\u094d\u0924\u0930 \u0936\u094d\u0930\u0947\u0922\u093c\u0940 21,18,15,... \u0915\u093e \u0915\u094c\u0928-\u0938\u093e \u092a\u0926 -81 \u0939\u0948? \u0938\u093e\u0925 \u0939\u0940 \u0915\u094d\u092f\u093e 0 \u0907\u0938 \u0938\u092e\u093e\u0928\u094d\u0924\u0930 \u0936\u094d\u0930\u0947\u0922\u093c\u0940 \u0915\u093e \u090f\u0915 \u092a\u0926 \u0939\u0948? \u0905\u092a\u0928\u0947 \u0909\u0924\u094d\u0924\u0930 \u0915\u0947 \u0932\u093f\u090f \u0915\u093e\u0930\u0923 \u0926\u0940\u091c\u093f\u090f\u0964<\/b><br>\na=21 d=-3<br>\ntn=21+(n-1)(-3)=24-3n<br>\n-81=24-3n\u21923n=105\u2192n=35 \u2192 35\u0935\u093e\u0901 \u092a\u0926 -81<br>\n0=24-3n\u21923n=24\u2192n=8 \u2192 8\u0935\u093e\u0901 \u092a\u0926 0 \u0939\u093e\u0901 \u092a\u0926 \u0939\u0948<br>\n<span style=\"background:#e8f5e9;padding:3px 10px;border-radius:6px;font-weight:bold;\">\u0909\u0924\u094d\u0924\u0930 35\u0935\u093e\u0901 \u092a\u0926 -81, 0 8\u0935\u093e\u0901 \u092a\u0926 \u0939\u093e\u0901 \u0939\u0948 \u0915\u094d\u092f\u094b\u0902\u0915\u093f n=8 \u092a\u094d\u0930\u093e\u0915\u0943\u0924<\/span><br><br>\n\n<b>Q3. \u0938\u092e\u093e\u0928\u094d\u0924\u0930 \u0936\u094d\u0930\u0947\u0922\u093c\u0940 11,8,5,2,... \u0915\u093e n\u0935\u093e\u0901 \u092a\u0926 \u091c\u094d\u091e\u093e\u0924 \u0915\u0940\u091c\u093f\u090f\u0964 \u0907\u0938 \u0936\u094d\u0930\u0947\u0922\u093c\u0940 \u0915\u0947 \u0932\u093f\u090f \u092a\u0941\u0928\u0930\u093e\u0935\u0930\u094d\u0924\u0940 \u0938\u0942\u0924\u094d\u0930 \u0932\u093f\u0916\u093f\u090f\u0964<\/b><br>\na=11 d=-3<br>\ntn=11+(n-1)(-3)=14-3n<br>\n\u092a\u0941\u0928\u0930\u093e\u0935\u0930\u094d\u0924\u0940: t1=11, tn=tn-1-3 n\u22652<br>\n<span style=\"background:#e8f5e9;padding:3px 10px;border-radius:6px;font-weight:bold;\">\u0909\u0924\u094d\u0924\u0930 tn=14-3n, t1=11 tn=tn-1-3<\/span><br>\n\n<\/div><\/div>\n\n<div id=\"q82-2\" style=\"display:none;\"><div style=\"background:#1565c0;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udd35 \u092d\u093e\u0917 2\/4 - Q4 Q5<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:14px;line-height:1.9;height:560px;overflow-y:auto;\">\n\n<b>Q4. \u090f\u0915 \u0938\u092e\u093e\u0928\u094d\u0924\u0930 \u0936\u094d\u0930\u0947\u0922\u093c\u0940 \u092e\u0947\u0902 50 \u092a\u0926 \u0939\u0948\u0902 \u092f\u0926\u093f \u0936\u094d\u0930\u0947\u0922\u093c\u0940 \u0915\u093e \u0924\u0940\u0938\u0930\u093e \u092a\u0926 12 \u0914\u0930 \u0905\u0902\u0924\u093f\u092e \u092a\u0926 106 \u0939\u0948 \u0924\u094b \u0907\u0938\u0915\u093e 29\u0935\u093e\u0901 \u092a\u0926 \u091c\u094d\u091e\u093e\u0924 \u0915\u0940\u091c\u093f\u090f\u0964 (\u0938\u0902\u0915\u0947\u0924\u2014 \u092f\u0926\u093f \u2018a\u2019 \u092a\u094d\u0930\u0925\u092e \u092a\u0926 \u0939\u094b \u0924\u0925\u093e \u2018d\u2019 \u0938\u093e\u0930\u094d\u0935 \u0905\u0902\u0924\u0930 \u0939\u094b, \u0924\u092c \u0939\u092e\u0947\u0902 \u0926\u094b \u0938\u092e\u0940\u0915\u0930\u0923 a+2d=12 \u0914\u0930 a+49d=106 \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0939\u094b\u0924\u093e \u0939\u0948\u0964 \u0930\u0948\u0916\u093f\u0915 \u0938\u092e\u0940\u0915\u0930\u0923\u094b\u0902 \u0915\u0947 \u0907\u0938 \u092f\u0941\u0917\u094d\u092e \u0915\u094b \u2018a\u2019 \u0924\u0925\u093e \u2018d\u2019 \u0915\u0947 \u0932\u093f\u090f \u0939\u0932 \u0915\u0940\u091c\u093f\u090f)\u0964<\/b><br>\na+2d=12...(1)<br>\na+49d=106...(2)<br>\n(2)-(1): 47d=94\u2192d=2<br>\na=12-4=8<br>\nt29=a+28d=8+56=64<br>\n<span style=\"background:#e8f5e9;padding:3px 10px;border-radius:6px;font-weight:bold;\">\u0909\u0924\u094d\u0924\u0930 64<\/span><br><br>\n\n<b>Q5. \u0915\u093f\u0924\u0928\u0940 2-\u0905\u0902\u0915\u0940\u092f \u0938\u0902\u0916\u094d\u092f\u093e\u090f\u0901 3 \u0938\u0947 \u0935\u093f\u092d\u093e\u091c\u094d\u092f \u0939\u0948\u0902? \u0907\u0928 \u0938\u092d\u0940 2-\u0905\u0902\u0915\u0940\u092f \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u093e \u092f\u094b\u0917\u092b\u0932 \u0915\u094d\u092f\u093e \u0939\u0948?<\/b><br>\n2-\u0905\u0902\u0915\u0940\u092f 10 \u0938\u0947 99, 3 \u0938\u0947 \u0935\u093f\u092d\u093e\u091c\u094d\u092f \u092a\u0939\u0932\u0940 12 \u0905\u0902\u0924\u093f\u092e 99<br>\nAP 12,15,18,...,99<br>\na=12 d=3 tn=99=12+(n-1)3\u21923n-3=87\u2192n=30 \u2192 30 \u0938\u0902\u0916\u094d\u092f\u093e\u090f\u0901<br>\n\u092f\u094b\u0917 Sn=n\/2(a+l)=30\/2(12+99)=15\u00d7111=1665<br>\n<span style=\"background:#e8f5e9;padding:3px 10px;border-radius:6px;font-weight:bold;\">\u0909\u0924\u094d\u0924\u0930 30 \u0938\u0902\u0916\u094d\u092f\u093e\u090f\u0901 \u092f\u094b\u0917 1665<\/span><br>\n\n<\/div><\/div>\n\n<div id=\"q82-3\" style=\"display:none;\"><div style=\"background:#2e7d32;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udfe2 \u092d\u093e\u0917 3\/4 - Q6 Q7<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:14px;line-height:1.9;height:560px;overflow-y:auto;\">\n\n<b>Q6. \u0939\u0930\u0940\u0936 \u0928\u0947 \u0915\u094b\u0908 \u0915\u093e\u0930\u094d\u092f (\u0928\u094c\u0915\u0930\u0940) \u20b95,00,000 \u0935\u093e\u0930\u094d\u0937\u093f\u0915 \u0906\u092f \u0938\u0947 \u092a\u094d\u0930\u093e\u0930\u0902\u092d \u0915\u0940 \u0914\u0930 \u092a\u094d\u0930\u0924\u093f\u0935\u0930\u094d\u0937 \u20b920,000 \u0935\u0947\u0924\u0928-\u0935\u0943\u0926\u094d\u0927\u093f \u0915\u0947 \u0930\u0942\u092a \u092e\u0947\u0902 \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0915\u093f\u090f\u0964 \u0915\u093f\u0924\u0928\u0947 \u0935\u0930\u094d\u0937\u094b\u0902 \u0915\u0947 \u092c\u093e\u0926 \u0909\u0938\u0915\u0940 \u0906\u092e\u0926\u0928\u0940 \u20b97,00,000 \u0939\u0941\u0908 \u0939\u094b\u0917\u0940?<\/b><br>\nAP a=5,00,000 d=20,000<br>\ntn=5,00,000+(n-1)20,000=7,00,000<br>\n(n-1)20,000=2,00,000\u2192n-1=10\u2192n=11<br>\n<span style=\"background:#e8f5e9;padding:3px 10px;border-radius:6px;font-weight:bold;\">\u0909\u0924\u094d\u0924\u0930 11\u0935\u0947\u0902 \u0935\u0930\u094d\u0937 \u092e\u0947\u0902 (10 \u0935\u0930\u094d\u0937 \u092c\u093e\u0926) 7 \u0932\u093e\u0916<\/span><br><br>\n\n<b>Q7. \u090f\u0915 \u092c\u091a\u094d\u091a\u093e \u0915\u0902\u091a\u094b\u0902 \u0915\u094b \u092a\u0902\u0915\u094d\u0924\u093f\u092f\u094b\u0902 \u092e\u0947\u0902 \u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930 \u0935\u094d\u092f\u0935\u0938\u094d\u0925\u093f\u0924 \u0915\u0930\u0924\u093e \u0939\u0948 \u0915\u093f \u092a\u0939\u0932\u0940 \u092a\u0902\u0915\u094d\u0924\u093f \u092e\u0947\u0902 1 \u0915\u0902\u091a\u093e \u0939\u0948, \u0926\u0942\u0938\u0930\u0940 \u092a\u0902\u0915\u094d\u0924\u093f \u092e\u0947\u0902 2 \u0915\u0902\u091a\u0947 \u0939\u0948\u0902, \u0924\u0940\u0938\u0930\u0940 \u092a\u0902\u0915\u094d\u0924\u093f \u092e\u0947\u0902 3 \u0915\u0902\u091a\u0947 \u0939\u0948\u0902 \u0924\u0925\u093e \u0907\u0938\u0940 \u092a\u094d\u0930\u0915\u093e\u0930 \u0938\u0947 25\u0935\u0940\u0902 \u092a\u0902\u0915\u094d\u0924\u093f \u0924\u0915 \u0915\u0902\u091a\u0947 \u0930\u0916\u0947 \u0917\u090f \u0939\u0948\u0902\u0964 \u092c\u091a\u094d\u091a\u093e \u0907\u0928\u092e\u0947\u0902 \u0915\u0941\u0932 \u0915\u093f\u0924\u0928\u0947 \u0915\u0902\u091a\u094b\u0902 \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0915\u0930\u0924\u093e \u0939\u0948?<\/b><br>\n1+2+3+...+25 = S25 =25\u00d726\/2=325<br>\n<span style=\"background:#e8f5e9;padding:3px 10px;border-radius:6px;font-weight:bold;\">\u0909\u0924\u094d\u0924\u0930 325 \u0915\u0902\u091a\u0947 - \u0924\u094d\u0930\u093f\u092d\u0941\u091c\u093e\u0915\u093e\u0930 25\u0935\u0940\u0902 \u0938\u0902\u0916\u094d\u092f\u093e<\/span><br>\n\n<\/div><\/div>\n\n<div id=\"q82-4\" style=\"display:none;\"><div style=\"background:#6a1b9a;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udfe3 \u092d\u093e\u0917 4\/4 - \u0938\u093e\u0930\u093e\u0902\u0936<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:14px;line-height:2;height:560px;overflow-y:auto;\">\n\n<div style=\"background:#fff9c4;border:3px solid #fbc02d;padding:10px;border-radius:10px;text-align:center;font-weight:bold;\">\n\u2705 \u092a\u094d\u0930\u0936\u094d\u0928\u093e\u0935\u0932\u0940 8.2 Q1-Q7 \u0936\u0941\u0926\u094d\u0927 \u0916\u0924\u094d\u092e<br><br>\nQ1 3,8,13... d5 t10=48 t26=128<br>\nQ2 21,18,15... d-3 35\u0935\u093e\u0901 -81, 0 8\u0935\u093e\u0901 \u0939\u093e\u0901<br>\nQ3 11,8,5,2... tn=14-3n \u092a\u0941\u0928\u0930\u093e\u0935\u0930\u094d\u0924\u0940 t1=11 tn=tn-1-3<br>\nQ4 50 \u092a\u0926 \u0924\u0940\u0938\u0930\u093e 12 \u0905\u0902\u0924\u093f\u092e 106 a8 d2 t29=64<br>\nQ5 2-\u0905\u0902\u0915\u0940\u092f 3 \u0938\u0947 \u0935\u093f\u092d\u093e\u091c\u094d\u092f 30 \u0938\u0902\u0916\u094d\u092f\u093e 12..99 \u092f\u094b\u0917 1665<br>\nQ6 \u0928\u094c\u0915\u0930\u0940 5 \u0932\u093e\u0916 d20k 11\u0935\u0947\u0902 \u0935\u0930\u094d\u0937 7 \u0932\u093e\u0916<br>\nQ7 1..25 \u0915\u0902\u091a\u0947 325 S25\n<\/div><br>\n\n<b>\u0938\u0942\u0924\u094d\u0930 \u092f\u093e\u0926:<\/b><br>\nAP tn=a+(n-1)d<br>\nSn=n\/2(a+l)<br>\nn\u0935\u093e\u0901 \u092a\u0926 3n-1 \u0906\u0926\u093f<br>\n\n<\/div><\/div>\n<\/div>\n\n<div style=\"background:#0a5c4b;padding:12px;display:flex;justify-content:space-between;align-items:center;\">\n<button type=\"button\" id=\"q82p\" style=\"background:#ffcc00;color:#000;border:3px solid #fff;border-radius:30px;padding:10px 22px;font-weight:bold;cursor:pointer;min-width:100px;\">&larr; \u092a\u0940\u091b\u0947<\/button>\n<span id=\"q82i\" style=\"background:#fff;color:#0a5c4b;padding:8px 20px;border-radius:20px;font-weight:bold;\">1 \/ 4<\/span>\n<button type=\"button\" id=\"q82n\" style=\"background:#00e676;color:#000;border:3px solid #fff;border-radius:30px;padding:10px 22px;font-weight:bold;cursor:pointer;min-width:100px;\">\u0906\u0917\u0947 &rarr;<\/button>\n<\/div>\n<\/div>\n\n<script>\n(function(){\nvar c=1,t=4;\nfunction s(n){\nfor(var i=1;i<=t;i++){\nvar el=document.getElementById('q82-'+i);\nif(el)el.style.display=(i===n?'block':'none');\n}\ndocument.getElementById('q82i').textContent=n+' \/ '+t;\nc=n;\n}\ndocument.getElementById('q82n').onclick=function(){var n=c+1;if(n>t)n=1;s(n);};\ndocument.getElementById('q82p').onclick=function(){var n=c-1;if(n<1)n=t;s(n);};\n})();\n<\/script><\/div><\/div><\/div><\/div><div id=\"pg-991-5\"  class=\"panel-grid panel-no-style\" ><div id=\"pgc-991-5-0\"  class=\"panel-grid-cell\" ><div id=\"panel-991-5-0-0\" class=\"widget_text so-panel widget widget_custom_html panel-first-child panel-last-child\" data-index=\"5\" ><div class=\"textwidget custom-html-widget\"><div id=\"CH86-GP\" style=\"max-width:860px;margin:12px auto;border:4px solid #0f5e4b;border-radius:14px;background:#e6f4f1;font-family:'Noto Sans Devanagari',Arial,sans-serif;overflow:hidden;\">\n<div style=\"background:#0a5c4b;padding:10px;text-align:center;\"><span style=\"background:#fff;color:#0a3d33;padding:7px 20px;border-radius:20px;font-weight:bold;font-size:13px;\">8.6 \u0917\u0941\u0923\u094b\u0924\u094d\u0924\u0930 \u0936\u094d\u0930\u0947\u0922\u093c\u0940 - \u092a\u0947\u091c 187-193 \u0936\u0941\u0926\u094d\u0927<\/span><\/div>\n\n<div style=\"padding:10px 12px;\">\n<div id=\"g86-1\" style=\"display:block;\"><div style=\"background:#d32f2f;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udd34 \u092d\u093e\u0917 1\/6 - GP \u092a\u0930\u093f\u091a\u092f<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:14px;line-height:2;height:560px;overflow-y:auto;\">\n\n<div style=\"background:#ffebee;border-left:4px solid #d32f2f;padding:8px 10px;font-weight:bold;\">8.6 \u0917\u0941\u0923\u094b\u0924\u094d\u0924\u0930 \u0936\u094d\u0930\u0947\u0922\u093c\u093f\u092f\u093e\u0901 \u0936\u0941\u0926\u094d\u0927<\/div>\n\u0907\u0938 \u0905\u0928\u0941\u092d\u093e\u0917 \u092e\u0947\u0902, \u0939\u092e \u090f\u0915 \u0905\u0928\u094d\u092f \u0935\u093f\u0936\u0947\u0937 \u092a\u094d\u0930\u0915\u093e\u0930 \u0915\u0947 \u0905\u0928\u0941\u0915\u094d\u0930\u092e \u0915\u093e \u0905\u0928\u094d\u0935\u0947\u0937\u0923 \u0915\u0930\u0947\u0902\u0917\u0947, \u091c\u094b \u0917\u0941\u0923\u094b\u0924\u094d\u0924\u0930 \u0936\u094d\u0930\u0947\u0922\u093c\u0940 \u0915\u0939\u0932\u093e\u0924\u093e \u0939\u0948\u0964 \u091a\u093f\u0924\u094d\u0930 8.6 \u092e\u0947\u0902 \u092a\u094d\u0930\u0926\u0930\u094d\u0936\u093f\u0924 \u0935\u0930\u094d\u0917\u094b\u0902 \u0915\u0947 \u092c\u0922\u093c\u0924\u0947 \u092a\u094d\u0930\u0924\u093f\u0930\u0942\u092a \u092a\u0930 \u0935\u093f\u091a\u093e\u0930 \u0915\u0940\u091c\u093f\u090f\u0964 \u091a\u093f\u0924\u094d\u0930 \u092e\u0947\u0902 \u092a\u094d\u0930\u0924\u093f\u0930\u0942\u092a \u0915\u0947 \u092a\u094d\u0930\u0925\u092e \u091a\u093e\u0930 \u091a\u0930\u0923 \u092a\u094d\u0930\u0926\u0930\u094d\u0936\u093f\u0924 \u0915\u093f\u090f \u0917\u090f \u0939\u0948\u0902\u0964 \u092f\u0926\u093f \u0939\u092e \u092a\u094d\u0930\u0924\u093f\u0930\u0942\u092a \u0915\u0947 \u091a\u093e\u0930 \u091a\u0930\u0923\u094b\u0902 \u092e\u0947\u0902 \u0939\u0930\u0947 \u0935\u0930\u094d\u0917\u094b\u0902 \u0915\u0940 \u0915\u0941\u0932 \u0938\u0902\u0916\u094d\u092f\u093e \u0917\u093f\u0928\u0924\u0947 \u0939\u0948\u0902, \u0924\u094b \u0939\u092e\u0947\u0902 \u0905\u0928\u0941\u0915\u094d\u0930\u092e 3,6,12,24 \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0939\u094b\u0924\u093e \u0939\u0948\u0964<br><br>\n\n\u0939\u092e \u0926\u0947\u0916\u0924\u0947 \u0939\u0948\u0902 \u0915\u093f \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u091a\u0930\u0923 \u092e\u0947\u0902, \u0935\u0930\u094d\u0917\u094b\u0902 \u0915\u0940 \u0938\u0902\u0916\u094d\u092f\u093e \u092a\u093f\u091b\u0932\u0947 \u091a\u0930\u0923 \u092e\u0947\u0902 \u0935\u0930\u094d\u0917\u094b\u0902 \u0915\u0940 \u0938\u0902\u0916\u094d\u092f\u093e \u0915\u0940 \u0926\u094b\u0917\u0941\u0928\u0940 \u0939\u0948\u0964 \u0906\u0917\u093e\u092e\u0940 \u091a\u0930\u0923\u094b\u0902 \u092e\u0947\u0902 \u0935\u0930\u094d\u0917\u094b\u0902 \u0915\u0940 \u0938\u0902\u0916\u094d\u092f\u093e 3, 3+3=6, 6+6=12, 12+12=24,... \u0939\u0948\u0964<br>\n\u0907\u0938\u0947 \u092a\u0941\u0928: 3, 3\u00d72, 3\u00d74, 3\u00d78,... \u092f\u093e 3, 3\u00d72, 3\u00d72\u00b2, 3\u00d72\u00b3,...<br>\nn\u0935\u093e\u0901 \u092a\u0926 tn=3\u00d72^(n-1)<br>\n\u092a\u0941\u0928\u0930\u093e\u0935\u0930\u094d\u0924\u0940 t1=3, tn=2tn-1 n\u22652<br>\n\u092a\u094d\u0930\u0925\u092e \u091b: \u092a\u0926 3,6,12,24,48,96<br><br>\n\n<div style=\"background:#fff9c4;border:2px solid #fbc02d;padding:8px;border-radius:8px;text-align:center;font-weight:bold;\">\n\u0905\u0928\u0941\u092a\u093e\u0924 6\/3=12\/6=24\/12=2 \u0938\u092e\u093e\u0928<br>\n\u0938\u093e\u0930\u094d\u0935 \u0905\u0928\u0941\u092a\u093e\u0924 r=2<br>\n\u0910\u0938\u093e \u0905\u0928\u0941\u0915\u094d\u0930\u092e \u0917\u0941\u0923\u094b\u0924\u094d\u0924\u0930 \u0936\u094d\u0930\u0947\u0922\u093c\u0940 (GP) \u0915\u0939\u0932\u093e\u0924\u093e \u0939\u0948\u0964\n<\/div><br>\n\n<div style=\"background:#e3f2fd;border:3px solid #1565c0;padding:10px;border-radius:10px;text-align:center;font-weight:bold;\">\n\u0935\u094d\u092f\u093e\u092a\u0915 GP: a, ar, ar\u00b2, ar\u00b3,... ar^(n-1)<br>\na \u092a\u094d\u0930\u0925\u092e \u092a\u0926, r \u0938\u093e\u0930\u094d\u0935 \u0905\u0928\u0941\u092a\u093e\u0924, tn=ar^(n-1)\n<\/div>\n\n<\/div><\/div>\n\n<div id=\"g86-2\" style=\"display:none;\"><div style=\"background:#1565c0;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udd35 \u092d\u093e\u0917 2\/6 - \u0909\u0926\u093e\u0939\u0930\u0923 6-9<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:14px;line-height:1.9;height:560px;overflow-y:auto;\">\n\n<b>\u0909\u0926\u093e\u0939\u0930\u0923 6\u2014 \u0915\u094d\u092f\u093e 1,2,4,8,16,... \u090f\u0915 GP \u0939\u0948? \u092f\u0926\u093f \u0939\u093e\u0901 \u0924\u094b r \u0915\u094d\u092f\u093e?<\/b><br>\n\u0939\u093e\u0901 r=2, a=1 tn=2^(n-1)<br><br>\n\n<b>\u0909\u0926\u093e\u0939\u0930\u0923 7\u2014 1,3,9,27,81,... GP?<\/b><br>\n\u0939\u093e\u0901 r=3 tn=3^(n-1)<br><br>\n\n<b>\u0909\u0926\u093e\u0939\u0930\u0923 8\u2014 1,-1,1,-1,1,... GP?<\/b><br>\n\u0939\u093e\u0901 r=-1 tn=(-1)^(n-1)<br><br>\n\n<b>\u0909\u0926\u093e\u0939\u0930\u0923 9 \u0936\u0941\u0926\u094d\u0927:<\/b> \u091c\u093e\u0901\u091a \u0915\u0940\u091c\u093f\u090f \u0915\u094d\u092f\u093e \u0905\u0928\u0941\u0915\u094d\u0930\u092e 5,15\/4,45\/16,135\/64,... \u090f\u0915 GP \u0939\u0948\u0964 \u0907\u0938\u0915\u093e n\u0935\u093e\u0901 \u092a\u0926 \u091c\u094d\u091e\u093e\u0924 \u0915\u0940\u091c\u093f\u090f\u0964<br>\nt1=5, t2=15\/4, t3=45\/16, t4=135\/64<br>\nt2\/t1=15\/4\u00f75=3\/4<br>\nt3\/t2=45\/16\u00f715\/4=3\/4<br>\nt4\/t3=135\/64\u00f745\/16=3\/4 \u2192 \u0905\u0928\u0941\u092a\u093e\u0924 \u0905\u091a\u0930 3\/4<br>\n\u0905\u0924\u0903 GP a=5 r=3\/4, n\u0935\u093e\u0901 \u092a\u0926=5\u00d7(3\/4)^(n-1)<br><br>\n\n<b>\u092a\u094d\u0930\u0936\u094d\u0928:<\/b> \u091c\u093e\u0901\u091a\u093f\u090f \u0915\u094d\u092f\u093e \u0928\u0940\u091a\u0947 \u0926\u093f\u090f \u0917\u090f \u0905\u0928\u0941\u0915\u094d\u0930\u092e GP \u0939\u0948\u0902 \u0914\u0930 \u0909\u0928\u0915\u0947 n\u0935\u0947\u0902 \u092a\u0926 \u091c\u094d\u091e\u093e\u0924 \u0915\u0940\u091c\u093f\u090f\u0964<br>\n(i) 2,10,50,250,... r=5 tn=2\u00d75^(n-1) \u0939\u093e\u0901 GP<br>\n(ii) 4,8\/3,16\/9,32\/27,... r=2\/3 tn=4\u00d7(2\/3)^(n-1) \u0939\u093e\u0901<br>\n(iii) 3,-3\/2,3\/4,-3\/8,... r=-1\/2 tn=3\u00d7(-1\/2)^(n-1) \u0939\u093e\u0901<br><br>\n\n<b>\u092a\u094d\u0930\u0936\u094d\u0928:<\/b> \u0915\u094d\u092f\u093e \u0906\u092a \u0938\u0942\u0924\u094d\u0930 tn=3\u00d710^(n-1) \u0915\u0947 \u0932\u093f\u090f \u092a\u0941\u0928\u0930\u093e\u0935\u0930\u094d\u0924\u0940 \u0938\u0942\u0924\u094d\u0930 \u0915\u093e \u092a\u0924\u093e \u0932\u0917\u093e \u0938\u0915\u0924\u0947 \u0939\u0948\u0902, \u091c\u093f\u0938\u0938\u0947 GP 3,30,300,3000,... \u092c\u0928\u0924\u0940 \u0939\u0948?<br>\nt1=3, tn=10\u00d7tn-1 n\u22652<br>\n\n<\/div><\/div>\n\n<div id=\"g86-3\" style=\"display:none;\"><div style=\"background:#2e7d32;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udfe2 \u092d\u093e\u0917 3\/6 - \u092b\u094d\u0930\u0948\u0915\u094d\u091f\u0932 \u0938\u093f\u0930\u092a\u093f\u0928\u094d\u0938\u094d\u0915\u0940<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:14px;line-height:1.9;height:560px;overflow-y:auto;\">\n\n<div style=\"background:#e8f5e9;border-left:4px solid #2e7d32;padding:8px 10px;font-weight:bold;\">8.6.1 \u092b\u094d\u0930\u0948\u0915\u094d\u091f\u0932\u094d\u0938 \u0915\u0947 \u0938\u093e\u0925 \u0906\u0928\u0902\u0926 \u0936\u0941\u0926\u094d\u0927<\/div>\n\u0915\u0915\u094d\u0937\u093e 8 \u0915\u0940 \u0917\u0923\u093f\u0924 \u092a\u093e\u0920\u094d\u092f\u092a\u0941\u0938\u094d\u0924\u0915 \u0938\u0947 \u0905\u0928\u094d\u092f \u0930\u094b\u091a\u0915 \u092a\u094d\u0930\u0924\u093f\u0930\u0942\u092a \u0915\u093e \u0938\u094d\u092e\u0930\u0923 \u0915\u0940\u091c\u093f\u090f \u091c\u0948\u0938\u093e \u0915\u093f \u091a\u093f\u0924\u094d\u0930 8.7 \u092e\u0947\u0902 \u0926\u0930\u094d\u0936\u093e\u092f\u093e \u0917\u092f\u093e \u0939\u0948\u0964 \u091a\u0930\u0923 0 \u0938\u092e\u092c\u093e\u0939\u0941 \u0924\u094d\u0930\u093f\u092d\u0941\u091c \u0915\u0947 \u0906\u0915\u093e\u0930 \u092e\u0947\u0902 \u0915\u093e\u091f\u0947 \u0917\u090f \u090f\u0915 \u0915\u093e\u0917\u091c \u0915\u0947 \u091f\u0941\u0915\u0921\u093c\u0947 \u0915\u094b \u0926\u0930\u094d\u0936\u093e\u0924\u093e \u0939\u0948\u0964 \u0906\u0907\u090f \u0924\u094d\u0930\u093f\u092d\u0941\u091c \u0915\u0940 \u0924\u0940\u0928\u094b\u0902 \u092d\u0941\u091c\u093e\u0913\u0902 \u0915\u0947 \u092e\u0927\u094d\u092f \u092c\u093f\u0902\u0926\u0941\u0913\u0902 \u0915\u094b \u092e\u093f\u0932\u093e\u0924\u0947 \u0939\u0948\u0902, \u091c\u093f\u0938\u0938\u0947 \u091a\u093e\u0930 \u091b\u094b\u091f\u0947 \u0938\u092e\u092c\u093e\u0939\u0941 \u0924\u094d\u0930\u093f\u092d\u0941\u091c \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0939\u094b\u0924\u0947 \u0939\u0948\u0902\u0964 \u0905\u092c \u092e\u0927\u094d\u092f \u092e\u0947\u0902 \u0938\u094d\u0925\u093f\u0924 \u0924\u094d\u0930\u093f\u092d\u0941\u091c \u0915\u094b \u0939\u091f\u093e \u0926\u0940\u091c\u093f\u090f\u0964 \u0906\u092a\u0915\u094b \u091a\u0930\u0923 1 \u092e\u0947\u0902 \u0926\u0930\u094d\u0936\u093e\u092f\u093e \u091a\u093f\u0924\u094d\u0930 \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0939\u094b\u0917\u093e \u091c\u093f\u0938\u0915\u0947 \u092e\u0927\u094d\u092f \u092e\u0947\u0902 \u0924\u094d\u0930\u093f\u092d\u0941\u091c\u093e\u0915\u093e\u0930 \u0930\u093f\u0915\u094d\u0924 \u0939\u094b\u0917\u0940\u0964 \u0907\u0938\u0940 \u092a\u094d\u0930\u0915\u094d\u0930\u093f\u092f\u093e \u0915\u094b \u091a\u0930\u0923 1 \u092e\u0947\u0902 \u0938\u092d\u0940 \u0924\u0940\u0928 \u0915\u093e\u0932\u0947 \u0924\u094d\u0930\u093f\u092d\u0941\u091c\u094b\u0902 \u092a\u0930 \u0926\u094b\u0939\u0930\u093e\u0907\u090f\u0964 \u0907\u0938\u0938\u0947 \u0906\u092a\u0915\u094b \u091a\u0930\u0923 2 \u092e\u0947\u0902 \u092a\u094d\u0930\u0926\u0930\u094d\u0936\u093f\u0924 \u091a\u093f\u0924\u094d\u0930 \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0939\u094b\u0917\u093e\u0964 \u0907\u0938\u0940 \u0935\u093f\u0927\u093f \u0938\u0947 \u0939\u092e \u091a\u0930\u0923 2 \u0938\u0947 \u091a\u0930\u0923 3 \u0924\u0915 \u092a\u0939\u0941\u0901\u091a\u0924\u0947 \u0939\u0948\u0902\u0964 \u0907\u0938 \u092a\u094d\u0930\u0915\u094d\u0930\u093f\u092f\u093e \u0915\u094b \u0928\u093f\u0930\u0902\u0924\u0930 \u091c\u093e\u0930\u0940 \u0930\u0916\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948\u0964 \u0907\u0938 \u0906\u0915\u0930\u094d\u0937\u0915 \u092a\u094d\u0930\u0924\u093f\u0930\u0942\u092a \u0915\u094b \u0938\u093f\u0930\u092a\u093f\u0928\u094d\u0938\u094d\u0915\u0940 \u0924\u094d\u0930\u093f\u092d\u0941\u091c \u0915\u0939\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964 \u0935\u093e\u0938\u094d\u0924\u0935 \u092e\u0947\u0902 \u092f\u0939 \u090f\u0915 \u092b\u094d\u0930\u0948\u0915\u094d\u091f\u0932 \u0939\u0948\u0964<br><br>\n\n\u0935\u093e\u0915\u094d\u0932\u093e\u0935 \u0938\u093f\u0930\u092a\u093f\u0928\u094d\u0938\u094d\u0915\u0940 (1882\u20131969) \u090f\u0915 \u092a\u094b\u0932\u093f\u0936 \u0917\u0923\u093f\u0924\u091c\u094d\u091e, \u091c\u093f\u0928\u094d\u0939\u0947\u0902 \u0917\u0923\u093f\u0924 \u092e\u0947\u0902 \u0909\u0928\u0915\u0947 \u0905\u0928\u0947\u0915 \u092f\u094b\u0917\u0926\u093e\u0928\u094b\u0902 \u0915\u0947 \u0932\u093f\u090f \u091c\u093e\u0928\u093e \u091c\u093e\u0924\u093e \u0939\u0948, \u091c\u093f\u0938\u092e\u0947\u0902 \u0938\u093f\u0930\u092a\u093f\u0928\u094d\u0938\u094d\u0915\u0940 \u0917\u093e\u0938\u094d\u0915\u0947\u091f \u092d\u0940 \u0938\u092e\u094d\u092e\u093f\u0932\u093f\u0924 \u0939\u0948, \u091c\u094b \u092b\u094d\u0930\u0948\u0915\u094d\u091f\u0932\u094d\u0938 \u0915\u0947 \u0938\u092c\u0938\u0947 \u092a\u094d\u0930\u093e\u0930\u0902\u092d\u093f\u0915 \u0914\u0930 \u0938\u092c\u0938\u0947 \u092a\u094d\u0930\u0938\u093f\u0926\u094d\u0927 \u0909\u0926\u093e\u0939\u0930\u0923\u094b\u0902 \u092e\u0947\u0902 \u0938\u0947 \u090f\u0915 \u0939\u0948\u0964<br><br>\n\n<b>\u0938\u094b\u091a\u093f\u090f \u0914\u0930 \u0935\u093f\u091a\u093e\u0930 \u0915\u0940\u091c\u093f\u090f:<\/b><br>\n(\u0915) \u091a\u093f\u0924\u094d\u0930 8.7 \u092e\u0947\u0902 \u091a\u0930\u0923 0 \u0938\u0947 \u091a\u0930\u0923 3 \u0924\u0915 \u0915\u093f\u0924\u0928\u0947 \u0915\u093e\u0932\u0947 \u0924\u094d\u0930\u093f\u092d\u0941\u091c \u0939\u0948\u0902?<br>\n(\u0916) \u0915\u094d\u092f\u093e \u0906\u092a \u091a\u0930\u0923 4 \u0914\u0930 5 \u092e\u0947\u0902 \u0915\u093e\u0932\u0947 \u0924\u094d\u0930\u093f\u092d\u0941\u091c\u094b\u0902 \u0915\u0940 \u0938\u0902\u0916\u094d\u092f\u093e \u0915\u093e \u092a\u0942\u0930\u094d\u0935\u093e\u0928\u0941\u092e\u093e\u0928 \u0932\u0917\u093e \u0938\u0915\u0924\u0947 \u0939\u0948\u0902?<br>\n(\u0917) \u0915\u094d\u092f\u093e \u0906\u092a n\u0935\u0947\u0902 \u091a\u0930\u0923 \u092e\u0947\u0902 \u0915\u093e\u0932\u0947 \u0924\u094d\u0930\u093f\u092d\u0941\u091c\u094b\u0902 \u0915\u0940 \u0938\u0902\u0916\u094d\u092f\u093e \u0915\u0947 \u0932\u093f\u090f \u0938\u0942\u0924\u094d\u0930 \u091c\u094d\u091e\u093e\u0924 \u0915\u0930 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902?<br>\n(\u0918) \u092e\u093e\u0928 \u0932\u0940\u091c\u093f\u090f \u091a\u0930\u0923 0 \u092e\u0947\u0902 \u0924\u094d\u0930\u093f\u092d\u0941\u091c \u0915\u093e \u0915\u094d\u0937\u0947\u0924\u094d\u0930\u092b\u0932 1 \u0935\u0930\u094d\u0917 \u0907\u0915\u093e\u0908 \u0939\u0948\u0964 \u091a\u0930\u0923 1,2,3 \u092e\u0947\u0902 \u0915\u093e\u0932\u0947 \u0915\u094d\u0937\u0947\u0924\u094d\u0930 \u0915\u093e \u0915\u094d\u0937\u0947\u0924\u094d\u0930\u092b\u0932 \u0915\u094d\u092f\u093e \u0939\u0948? \u091a\u0930\u0923 4,5 \u092e\u0947\u0902? n\u0935\u0947\u0902 \u091a\u0930\u0923 \u092e\u0947\u0902?<br><br>\n\n\u0939\u0932 \u0936\u0941\u0926\u094d\u0927:<br>\n\u091a\u0930\u0923 0,1,2,3 \u092e\u0947\u0902 \u0915\u093e\u0932\u0947 \u0924\u094d\u0930\u093f\u092d\u0941\u091c\u094b\u0902 \u0915\u0940 \u0938\u0902\u0916\u094d\u092f\u093e \u0915\u094d\u0930\u092e\u0936\u0903 1,3,9,27 \u0939\u0948\u0964 \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u091a\u0930\u0923 \u092e\u0947\u0902 \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u0915\u093e\u0932\u093e \u0924\u094d\u0930\u093f\u092d\u0941\u091c \u0905\u0917\u0932\u0947 \u091a\u0930\u0923 \u092e\u0947\u0902 \u0924\u0940\u0928 \u091b\u094b\u091f\u0947 \u0924\u094d\u0930\u093f\u092d\u0941\u091c\u094b\u0902 \u0938\u0947 \u0935\u093f\u0938\u094d\u0925\u093e\u092a\u093f\u0924 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964 \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u091a\u0930\u0923 \u092e\u0947\u0902 \u0915\u093e\u0932\u0947 \u0924\u094d\u0930\u093f\u092d\u0941\u091c\u094b\u0902 \u0915\u0940 \u0938\u0902\u0916\u094d\u092f\u093e \u092a\u093f\u091b\u0932\u0947 \u091a\u0930\u0923 \u0938\u0947 \u0924\u0940\u0928 \u0917\u0941\u0928\u093e\u0964<br>\n\u091a\u0930\u0923 4=81, \u091a\u0930\u09235=243<br>\n\u0905\u0928\u0941\u0915\u094d\u0930\u092e 1,3,9,27,81,243,... GP r=3, \u0938\u092d\u0940 \u092a\u0926 3 \u0915\u0940 \u0918\u093e\u0924\u0947\u0902 3^0,3^1,3^2... \u0918\u093e\u0924 \u091a\u0930\u0923 \u0938\u0902\u0916\u094d\u092f\u093e \u0915\u0947 \u092c\u0930\u093e\u092c\u0930\u0964 n\u0935\u0947\u0902 \u091a\u0930\u0923 \u092e\u0947\u0902 \u0915\u093e\u0932\u0947 \u0924\u094d\u0930\u093f\u092d\u0941\u091c\u094b\u0902 \u0915\u0940 \u0938\u0902\u0916\u094d\u092f\u093e 3^n<br>\n\n<\/div><\/div>\n\n<div id=\"g86-4\" style=\"display:none;\"><div style=\"background:#6a1b9a;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udfe3 \u092d\u093e\u0917 4\/6 - \u0915\u094d\u0937\u0947\u0924\u094d\u0930\u092b\u0932 GP<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:14px;line-height:1.9;height:560px;overflow-y:auto;\">\n\n<b>\u0915\u094d\u0937\u0947\u0924\u094d\u0930\u092b\u0932 \u0936\u0941\u0926\u094d\u0927:<\/b><br>\n\u091a\u0930\u0923 0 \u092e\u0947\u0902 \u0915\u093e\u0932\u093e \u0915\u094d\u0937\u0947\u0924\u094d\u0930 1<br>\n\u091a\u0930\u0923 1 \u092e\u0947\u0902 \u0924\u094d\u0930\u093f\u092d\u0941\u091c 4 \u0938\u092e\u093e\u0928 \u092d\u093e\u0917\u094b\u0902 \u092e\u0947\u0902 \u092c\u093e\u0901\u091f\u093e \u092e\u0927\u094d\u092f \u0939\u091f\u093e \u2192 \u0915\u093e\u0932\u093e 3\/4<br>\n\u091a\u0930\u0923 2 \u092e\u0947\u0902 \u0915\u093e\u0932\u093e \u091a\u0930\u0923 1 \u0915\u093e 3\/4 \u092d\u093e\u0917 \u2192 3\/4\u00d73\/4=(3\/4)\u00b2<br>\nn\u0935\u0947\u0902 \u091a\u0930\u0923 \u092e\u0947\u0902 \u0915\u093e\u0932\u093e \u0915\u094d\u0937\u0947\u0924\u094d\u0930 (3\/4)^n<br><br>\n\n\u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u091a\u0930\u0923 \u092e\u0947\u0902 \u0915\u093e\u0932\u0947 \u0915\u094d\u0937\u0947\u0924\u094d\u0930 \u0915\u0947 \u0915\u094d\u0937\u0947\u0924\u094d\u0930\u092b\u0932 \u0938\u0947 GP \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0939\u094b\u0924\u0940 \u0939\u0948, \u091c\u093f\u0938\u092e\u0947\u0902 \u0905\u0928\u0941\u0915\u094d\u0930\u092e \u0915\u093e \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u0909\u0924\u094d\u0924\u0930\u094b\u0924\u094d\u0924\u0930 \u092a\u0926 \u092a\u093f\u091b\u0932\u0947 \u092a\u0926 \u0915\u094b 3\/4 \u0938\u0947 \u0917\u0941\u0923\u093e \u0915\u0930\u0928\u0947 \u092a\u0930 \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964 \u0915\u093e\u0932\u0947 \u0924\u094d\u0930\u093f\u092d\u0941\u091c\u094b\u0902 \u0915\u0940 \u0938\u0902\u0916\u094d\u092f\u093e \u0924\u0940\u0935\u094d\u0930\u0924\u093e \u0938\u0947 \u092c\u0922\u093c\u0928\u0947 \u0915\u0947 \u0938\u093e\u0925 \u0924\u094d\u0930\u093f\u092d\u0941\u091c\u094b\u0902 \u0915\u093e \u0915\u0941\u0932 \u0915\u094d\u0937\u0947\u0924\u094d\u0930\u092b\u0932 \u0918\u091f\u0924\u093e \u0939\u0948\u0964<br><br>\n\n<div style=\"background:#fff;border:2px solid #0f5e4b;border-radius:8px;padding:8px;\">\n<table style=\"width:100%;border-collapse:collapse;text-align:center;\">\n<tr style=\"background:#0a5c4b;color:#fff;\"><th style=\"border:1px solid #000;padding:4px;\">\u091a\u0930\u0923 n<\/th><th style=\"border:1px solid #000;padding:4px;\">0<\/th><th style=\"border:1px solid #000;padding:4px;\">1<\/th><th style=\"border:1px solid #000;padding:4px;\">2<\/th><th style=\"border:1px solid #000;padding:4px;\">3<\/th><th style=\"border:1px solid #000;padding:4px;\">4<\/th><th style=\"border:1px solid #000;padding:4px;\">5<\/th><th style=\"border:1px solid #000;padding:4px;\">n<\/th><\/tr>\n<tr><td style=\"border:1px solid #000;padding:4px;\">\u0915\u093e\u0932\u0947 \u0924\u094d\u0930\u093f\u092d\u0941\u091c tn<\/td><td style=\"border:1px solid #000;padding:4px;\">1=3^0<\/td><td style=\"border:1px solid #000;padding:4px;\">3=3^1<\/td><td style=\"border:1px solid #000;padding:4px;\">9=3\u00b2<\/td><td style=\"border:1px solid #000;padding:4px;\">27=3\u00b3<\/td><td style=\"border:1px solid #000;padding:4px;\">81=3^4<\/td><td style=\"border:1px solid #000;padding:4px;\">243=3^5<\/td><td style=\"border:1px solid #000;padding:4px;\">3^n<\/td><\/tr>\n<tr><td style=\"border:1px solid #000;padding:4px;\">\u091b\u093e\u092f\u093e\u0902\u0915\u093f\u0924 \u0915\u094d\u0937\u0947\u0924\u094d\u0930 sn<\/td><td style=\"border:1px solid #000;padding:4px;\">1<\/td><td style=\"border:1px solid #000;padding:4px;\">3\/4<\/td><td style=\"border:1px solid #000;padding:4px;\">(3\/4)\u00b2<\/td><td style=\"border:1px solid #000;padding:4px;\">(3\/4)\u00b3<\/td><td style=\"border:1px solid #000;padding:4px;\">(3\/4)^4<\/td><td style=\"border:1px solid #000;padding:4px;\">(3\/4)^5<\/td><td style=\"border:1px solid #000;padding:4px;\">(3\/4)^n<\/td><\/tr>\n<\/table>\n<\/div><br>\n\n\u0935\u094d\u092f\u093e\u092a\u0915 \u0938\u0942\u0924\u094d\u0930 tn=3^n \u0914\u0930 sn=(3\/4)^n<br>\n\u092a\u0941\u0928\u0930\u093e\u0935\u0930\u094d\u0924\u0940 t1=1 tn=3\u00d7tn-1 n\u22652 \u0914\u0930 s1=1 sn=3\/4\u00d7sn-1<br>\n\u092b\u094d\u0930\u0948\u0915\u094d\u091f\u0932\u094d\u0938 \u0935\u0947 \u0906\u0915\u0943\u0924\u093f\u092f\u093e\u0901 \u092f\u093e \u092a\u094d\u0930\u0924\u093f\u0930\u0942\u092a \u0939\u0948\u0902 \u091c\u093f\u0928\u0915\u093e \u0935\u0939\u0940 \u0930\u0942\u092a \u0935\u093f\u092d\u093f\u0928\u094d\u0928 \u0938\u094d\u0924\u0930\u094b\u0902 \u092a\u0930 \u092c\u093e\u0930-\u092c\u093e\u0930 \u0926\u093f\u0916\u093e\u0908 \u0926\u0947\u0924\u093e \u0939\u0948\u0964 \u092b\u094d\u0930\u0948\u0915\u094d\u091f\u0932\u094d\u0938 \u092a\u094d\u0930\u0915\u0943\u0924\u093f \u092e\u0947\u0902 \u0938\u092d\u0940 \u091c\u0917\u0939 \u092a\u093e\u090f \u091c\u093e\u0924\u0947 \u0939\u0948\u0902 - \u0935\u0943\u0915\u094d\u0937\u094b\u0902 \u0915\u0940 \u0936\u093e\u0916\u093e\u0913\u0902 \u092e\u0947\u0902, \u092b\u0942\u0932\u0917\u094b\u092d\u0940 \u092f\u093e \u092c\u094d\u0930\u094b\u0915\u094b\u0932\u0940 \u092e\u0947\u0902, \u0939\u093f\u092e\u0915\u0923\u094b\u0902 \u092e\u0947\u0902 \u092f\u093e \u0924\u091f\u0940\u092f \u0930\u0947\u0916\u093e\u0913\u0902 \u092e\u0947\u0902\u0964<br>\n\n<\/div><\/div>\n\n<div id=\"g86-5\" style=\"display:none;\"><div style=\"background:#ff6f00;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udfe0 \u092d\u093e\u0917 5\/6 - GP \u0926\u0943\u0936\u094d\u092f\u093e\u0902\u0915\u0928 \u0914\u0930 \u0909\u0926\u093e\u0939\u0930\u0923 10<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:14px;line-height:1.9;height:560px;overflow-y:auto;\">\n\n<div style=\"background:#fff3e0;border-left:4px solid #ff6f00;padding:8px 10px;font-weight:bold;\">8.6.2 GP \u0915\u093e \u0926\u0943\u0936\u094d\u092f\u093e\u0902\u0915\u0928 \u0936\u0941\u0926\u094d\u0927<\/div>\n\u091a\u093f\u0924\u094d\u0930 8.6 \u092e\u0947\u0902 \u0935\u0930\u094d\u0917\u094b\u0902 \u0915\u0947 \u092c\u0922\u093c\u0924\u0947 \u092a\u094d\u0930\u0924\u093f\u0930\u0942\u092a \u0915\u094b \u092a\u0941\u0928: \u0926\u0947\u0916\u093f\u090f - \u091a\u0930\u0923 1 2 3 4 5 ... n, \u0935\u0930\u094d\u0917\u094b\u0902 \u0915\u0940 \u0938\u0902\u0916\u094d\u092f\u093e 3 6 12 24 48 ... 3\u00d72^(n-1)<br>\n\u092f\u0941\u0917\u094d\u092e (1,3)(2,6)(3,12)(4,24)(5,48) \u0906\u0932\u0947\u0916 \u092a\u0930 \u0938\u0930\u0932 \u0930\u0947\u0916\u093e \u092a\u0930 \u0928\u0939\u0940\u0902 \u0939\u0948\u0902 - \u091a\u093f\u0924\u094d\u0930 8.9 GP \u092c\u093f\u0902\u0926\u0941 \u0938\u0930\u0932 \u0930\u0947\u0916\u093e \u092a\u0930 \u0928\u0939\u0940\u0902, \u0924\u0940\u0935\u094d\u0930 \u0935\u0943\u0926\u094d\u0927\u093f<br>\n\u0938\u093f\u0930\u092a\u093f\u0928\u094d\u0938\u094d\u0915\u0940 \u0924\u094d\u0930\u093f\u092d\u0941\u091c \u092e\u0947\u0902 \u0915\u093e\u0932\u0947 \u0924\u094d\u0930\u093f\u092d\u0941\u091c\u094b\u0902 \u0915\u0940 \u0938\u0902\u0916\u094d\u092f\u093e \u092c\u0939\u0941\u0924 \u0924\u0940\u0935\u094d\u0930\u0924\u093e \u0938\u0947 \u092c\u0922\u093c\u0924\u0940 \u0939\u0948 \u091c\u092c\u0915\u093f \u0915\u093e\u0932\u0947 \u0915\u094d\u0937\u0947\u0924\u094d\u0930 \u0915\u093e \u0915\u094d\u0937\u0947\u0924\u094d\u0930\u092b\u0932 \u0915\u092e \u0939\u094b\u0915\u0930 0 \u0915\u0947 \u0928\u093f\u0915\u091f - \u091a\u093f\u0924\u094d\u0930 8.10<br><br>\n\n<b>\u0909\u0926\u093e\u0939\u0930\u0923 10 \u0936\u0941\u0926\u094d\u0927:<\/b> \u090f\u0915 \u0917\u0947\u0902\u0926 \u092d\u0942\u092e\u093f \u0938\u0947 24 \u092b\u0941\u091f \u090a\u0901\u091a\u093e\u0908 \u0938\u0947 \u0917\u093f\u0930\u093e\u0908 \u091c\u093e\u0924\u0940 \u0939\u0948\u0964 \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u092c\u093e\u0930 \u0917\u0947\u0902\u0926 \u092d\u0942\u092e\u093f \u0938\u0947 \u091f\u0915\u0930\u093e\u0915\u0930 \u0907\u0938\u0915\u0940 \u092a\u093f\u091b\u0932\u0940 \u090a\u0901\u091a\u093e\u0908 \u0915\u0947 3\/4 \u092d\u093e\u0917 \u090a\u0901\u091a\u093e\u0908 \u0924\u0915 \u090a\u092a\u0930 \u091c\u093e\u0924\u0940 \u0939\u0948\u0964<br>\n(\u0915) \u0917\u0947\u0902\u0926 \u0915\u0947 \u092a\u093e\u0901\u091a \u0909\u091b\u093e\u0932\u094b\u0902 \u092e\u0947\u0902 \u0917\u0947\u0902\u0926 \u0915\u094b \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u090a\u0901\u091a\u093e\u0907\u092f\u094b\u0902 \u0915\u093e \u0905\u0928\u0941\u0915\u094d\u0930\u092e \u0932\u093f\u0916\u093f\u090f?<br>\n(\u0916) \u092a\u094d\u0930\u093e\u0930\u0902\u092d \u092e\u0947\u0902 \u0917\u0947\u0902\u0926 \u0915\u094b \u0917\u093f\u0930\u093e\u090f \u091c\u093e\u0928\u0947 \u0915\u0940 \u090a\u0901\u091a\u093e\u0908 \u0915\u0947 1\/6 \u092d\u093e\u0917 \u0938\u0947 \u0915\u092e \u090a\u0901\u091a\u093e\u0908 \u0924\u0915 \u0917\u0947\u0902\u0926 \u0915\u0947 \u092a\u0939\u0941\u0901\u091a\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u0915\u093f\u0924\u0928\u0947 \u092a\u094d\u0930\u092f\u093e\u0938\u094b\u0902 (\u0909\u091b\u093e\u0932\u094b\u0902) \u0915\u0940 \u0906\u0935\u0936\u094d\u092f\u0915\u0924\u093e?<br>\n\u0939\u0932:<br>\n\u092a\u094d\u0930\u0925\u092e \u0909\u091b\u093e\u0932 24\u00d70.75=18 \u092b\u0941\u091f<br>\n\u0926\u0942\u0938\u0930\u0940 18\u00d70.75=13.5<br>\n\u0924\u0940\u0938\u0930\u0940 13.5\u00d70.75=10.125<br>\n\u091a\u094c\u0925\u0940 10.125\u00d70.75=7.59375<br>\n\u092a\u093e\u0901\u091a\u0935\u0940\u0902 5.695, \u091b\u0920\u0940 4.27125, \u0938\u093e\u0924\u0935\u0940\u0902 3.2034375<br>\n\u090a\u0901\u091a\u093e\u0907\u092f\u094b\u0902 \u0915\u093e \u0905\u0928\u0941\u0915\u094d\u0930\u092e GP a=18 r=0.75=3\/4<br>\nGP 18,13.5,10.125,7.594,5.695,4.271,3.203...<br>\n\u092a\u094d\u0930\u093e\u0930\u0902\u092d\u093f\u0915 24 \u0915\u093e 1\/6=4 \u092b\u0941\u091f, 7\u0935\u0940\u0902 \u0909\u091b\u093e\u0932 \u0915\u0947 \u092c\u093e\u0926 3.20<4 \u2192 7 \u0909\u091b\u093e\u0932 \u092c\u093e\u0926 1\/6 \u0938\u0947 \u0915\u092e<br>\n<span style=\"background:#e8f5e9;padding:3px 10px;border-radius:6px;font-weight:bold;\">\u0909\u0924\u094d\u0924\u0930 7 \u0909\u091b\u093e\u0932<\/span><br>\n\n<\/div><\/div>\n\n<div id=\"g86-6\" style=\"display:none;\"><div style=\"background:#0f5e4b;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\u2705 \u092d\u093e\u0917 6\/6 - FINAL \u0938\u093e\u0930<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:14px;line-height:2;height:560px;overflow-y:auto;\">\n\n<div style=\"background:#fff9c4;border:3px solid #fbc02d;padding:10px;border-radius:10px;text-align:center;font-weight:bold;\">\n\u2705 8.6 GP FINAL \u0936\u0941\u0926\u094d\u0927<br><br>\nGP \u092a\u0930\u093f\u092d\u093e\u0937\u093e \u0905\u0928\u0941\u092a\u093e\u0924 \u0905\u091a\u0930 r<br>\na,ar,ar\u00b2... tn=ar^(n-1)<br>\n\u092a\u0941\u0928\u0930\u093e\u0935\u0930\u094d\u0924\u0940 t1=a tn=r\u00d7tn-1<br>\n\u0909\u0926\u093e 3,6,12,24 a3 r2 tn3\u00d72^(n-1)<br>\n1,2,4,8 r2, 1,3,9 r3, 1,-1,1 r-1<br>\n5,15\/4... a5 r3\/4 tn5\u00d7(3\/4)^(n-1)<br>\n2,10,50 r5, 4,8\/3 r2\/3, 3,-3\/2 r-1\/2<br>\n\u0938\u093f\u0930\u092a\u093f\u0928\u094d\u0938\u094d\u0915\u0940 \u0915\u093e\u0932\u0947 \u0924\u094d\u0930\u093f\u092d\u0941\u091c 1,3,9,27 GP r3 tn3^n \u0938\u0902\u0916\u094d\u092f\u093e \u0924\u0947\u091c\u0940 \u0938\u0947 \u092c\u0922\u093c\u0924\u0940<br>\n\u0915\u094d\u0937\u0947\u0924\u094d\u0930\u092b\u0932 1,3\/4,(3\/4)\u00b2... GP r3\/4 sn(3\/4)^n \u0918\u091f\u0915\u0930 0 \u0915\u0940 \u0913\u0930<br>\n\u092b\u094d\u0930\u0948\u0915\u094d\u091f\u0932 \u0938\u094d\u0935-\u0938\u092e\u093e\u0928 \u092a\u094d\u0930\u0915\u0943\u0924\u093f \u092e\u0947\u0902<br>\n\u0917\u0947\u0902\u0926 24 \u092b\u0941\u091f 3\/4 \u0909\u091b\u093e\u0932 GP 18,13.5... 7 \u0909\u091b\u093e\u0932 \u092c\u093e\u0926 1\/6 \u0938\u0947 \u0915\u092e<br>\nAP \u092c\u093f\u0902\u0926\u0941 \u0938\u0930\u0932 \u0930\u0947\u0916\u093e, GP \u092c\u093f\u0902\u0926\u0941 \u0938\u0930\u0932 \u0930\u0947\u0916\u093e \u0928\u0939\u0940\u0902 \u0924\u0947\u091c\u0940 \u0938\u0947 \u0935\u0943\u0926\u094d\u0927\u093f\n<\/div>\n\n<\/div><\/div>\n<\/div>\n\n<div style=\"background:#0a5c4b;padding:12px;display:flex;justify-content:space-between;align-items:center;\">\n<button type=\"button\" id=\"g86p\" style=\"background:#ffcc00;color:#000;border:3px solid #fff;border-radius:30px;padding:10px 22px;font-weight:bold;cursor:pointer;min-width:100px;\">&larr; \u092a\u0940\u091b\u0947<\/button>\n<span id=\"g86i\" style=\"background:#fff;color:#0a5c4b;padding:8px 20px;border-radius:20px;font-weight:bold;\">1 \/ 6<\/span>\n<button type=\"button\" id=\"g86n\" style=\"background:#00e676;color:#000;border:3px solid #fff;border-radius:30px;padding:10px 22px;font-weight:bold;cursor:pointer;min-width:100px;\">\u0906\u0917\u0947 &rarr;<\/button>\n<\/div>\n<\/div>\n\n<script>\n(function(){\nvar c=1,t=6;\nfunction s(n){\nfor(var i=1;i<=t;i++){\nvar el=document.getElementById('g86-'+i);\nif(el)el.style.display=(i===n?'block':'none');\n}\ndocument.getElementById('g86i').textContent=n+' \/ '+t;\nc=n;\n}\ndocument.getElementById('g86n').onclick=function(){var n=c+1;if(n>t)n=1;s(n);};\ndocument.getElementById('g86p').onclick=function(){var n=c-1;if(n<1)n=t;s(n);};\n})();\n<\/script><\/div><\/div><\/div><\/div><div id=\"pg-991-6\"  class=\"panel-grid panel-no-style\" ><div id=\"pgc-991-6-0\"  class=\"panel-grid-cell\" ><div id=\"panel-991-6-0-0\" class=\"widget_text so-panel widget widget_custom_html panel-first-child panel-last-child\" data-index=\"6\" ><div class=\"textwidget custom-html-widget\"><div id=\"Q83-ALL\" style=\"max-width:860px;margin:12px auto;border:4px solid #0f5e4b;border-radius:14px;background:#e6f4f1;font-family:'Noto Sans Devanagari',Arial,sans-serif;overflow:hidden;\">\n<div style=\"background:#0a5c4b;padding:10px;text-align:center;\"><span style=\"background:#fff;color:#0a3d33;padding:7px 20px;border-radius:20px;font-weight:bold;font-size:13px;\">\u092a\u094d\u0930\u0936\u094d\u0928\u093e\u0935\u0932\u0940 8.3 - Q1-Q7 - \u092a\u0947\u091c 194<\/span><\/div>\n\n<div style=\"padding:10px 12px;\">\n<div id=\"q83-1\" style=\"display:block;\"><div style=\"background:#d32f2f;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udd34 \u092d\u093e\u0917 1\/4 - Q1-Q4<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:14px;line-height:1.9;height:560px;overflow-y:auto;\">\n\n<b>Q1. \u0909\u0938 \u0917\u0941\u0923\u094b\u0924\u094d\u0924\u0930 \u0936\u094d\u0930\u0947\u0922\u093c\u0940 \u0915\u093e 12\u0935\u093e\u0901 \u092a\u0926 \u091c\u094d\u091e\u093e\u0924 \u0915\u0940\u091c\u093f\u090f, \u091c\u093f\u0938\u0915\u093e \u0938\u093e\u0930\u094d\u0935 \u0905\u0928\u0941\u092a\u093e\u0924 2 \u0939\u0948 \u0914\u0930 8\u0935\u093e\u0901 \u092a\u0926 192 \u0939\u0948\u0964<\/b><br>\nt8=ar^7=192 r=2\u2192a\u00d7128=192\u2192a=1.5<br>\nt12=ar^11=1.5\u00d72048=3072<br>\n<span style=\"background:#e8f5e9;padding:3px 10px;border-radius:6px;font-weight:bold;\">\u0909\u0924\u094d\u0924\u0930 3072<\/span><br><br>\n\n<b>Q2. \u0917\u0941\u0923\u094b\u0924\u094d\u0924\u0930 \u0936\u094d\u0930\u0947\u0922\u093c\u0940 5,25,125,... \u0915\u093e 10\u0935\u093e\u0901 \u092a\u0926 \u0924\u0925\u093e n\u0935\u093e\u0901 \u092a\u0926 \u091c\u094d\u091e\u093e\u0924 \u0915\u0940\u091c\u093f\u090f\u0964<\/b><br>\na=5 r=5 tn=5\u00d75^(n-1)=5^n<br>\nt10=5^10=9765625<br>\n<span style=\"background:#e8f5e9;padding:3px 10px;border-radius:6px;font-weight:bold;\">\u0909\u0924\u094d\u0924\u0930 t10=9765625 tn=5^n<\/span><br><br>\n\n<b>Q3. \u090f\u0915 \u0905\u0928\u0941\u0915\u094d\u0930\u092e \u0915\u093e \u092a\u0941\u0928\u0930\u093e\u0935\u0930\u094d\u0924\u0940 \u0938\u0942\u0924\u094d\u0930 t1=2, tn+1=3tn-2, \u091c\u092c\u0915\u093f n\u22651 \u0939\u0948\u0964 \u0907\u0938 \u0905\u0928\u0941\u0915\u094d\u0930\u092e \u0915\u093e \u0915\u094c\u0928-\u0938\u093e \u092a\u0926 730 \u0939\u0948?<\/b><br>\nt1=2<br>\nt2=3\u00d72-2=4<br>\nt3=3\u00d74-2=10<br>\nt4=28, t5=82, t6=244, t7=730<br>\n<span style=\"background:#e8f5e9;padding:3px 10px;border-radius:6px;font-weight:bold;\">\u0909\u0924\u094d\u0924\u0930 7\u0935\u093e\u0901 \u092a\u0926<\/span> \u092a\u0948\u091f\u0930\u094d\u0928 tn=3^(n-1)+1 \u2192 3^6+1=730<br><br>\n\n<b>Q4. \u0917\u0941\u0923\u094b\u0924\u094d\u0924\u0930 \u0936\u094d\u0930\u0947\u0922\u093c\u0940 2,6,18,... \u0915\u093e \u0915\u094c\u0928-\u0938\u093e \u092a\u0926 4374 \u0939\u0948? \u0907\u0938 \u0936\u094d\u0930\u0947\u0922\u093c\u0940 \u0915\u0947 n\u0935\u0947\u0902 \u092a\u0926 \u0915\u0947 \u0932\u093f\u090f \u092e\u093e\u0928\u0915 \u0938\u0942\u0924\u094d\u0930 \u0914\u0930 \u092a\u0941\u0928\u0930\u093e\u0935\u0930\u094d\u0924\u0940 \u0938\u0942\u0924\u094d\u0930 \u092d\u0940 \u0932\u093f\u0916\u093f\u090f\u0964<\/b><br>\na=2 r=3 tn=2\u00d73^(n-1)=4374\u21923^(n-1)=2187=3^7\u2192n-1=7\u2192n=8<br>\n\u092e\u093e\u0928\u0915 tn=2\u00d73^(n-1), \u092a\u0941\u0928\u0930\u093e\u0935\u0930\u094d\u0924\u0940 t1=2 tn=3tn-1<br>\n<span style=\"background:#e8f5e9;padding:3px 10px;border-radius:6px;font-weight:bold;\">\u0909\u0924\u094d\u0924\u0930 8\u0935\u093e\u0901 \u092a\u0926<\/span><br>\n\n<\/div><\/div>\n\n<div id=\"q83-2\" style=\"display:none;\"><div style=\"background:#1565c0;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udd35 \u092d\u093e\u0917 2\/4 - Q5 \u0917\u0947\u0902\u0926<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:14px;line-height:1.9;height:560px;overflow-y:auto;\">\n\n<b>Q5. \u090f\u0915 \u0917\u0947\u0902\u0926 \u0915\u094b 80 \u092e\u0940\u091f\u0930 \u0915\u0940 \u090a\u0901\u091a\u093e\u0908 \u0938\u0947 \u0917\u093f\u0930\u093e\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964 \u092d\u0942\u092e\u093f \u0938\u0947 \u091f\u0915\u0930\u093e\u0928\u0947 \u0915\u0947 \u092c\u093e\u0926 \u0917\u0947\u0902\u0926 \u0905\u092a\u0928\u0940 \u092a\u094d\u0930\u093e\u0930\u0902\u092d\u093f\u0915 \u090a\u0901\u091a\u093e\u0908 \u0915\u0947 60% \u0924\u0915 \u092d\u0942\u092e\u093f \u0915\u0947 \u090a\u092a\u0930 \u091c\u093e\u0924\u0940 \u0939\u0948\u0964 \u0907\u0938\u0940 \u092a\u094d\u0930\u0915\u093e\u0930 \u0938\u0947 \u0917\u0947\u0902\u0926 \u0915\u0940 \u0909\u091b\u093e\u0932 \u091c\u093e\u0930\u0940 \u0930\u0939\u0924\u0940 \u0939\u0948 \u0905\u0930\u094d\u0925\u093e\u0924 \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u092c\u093e\u0930 \u0917\u0947\u0902\u0926 \u092d\u0942\u092e\u093f \u0938\u0947 \u091f\u0915\u0930\u093e\u0915\u0930 \u0905\u092a\u0928\u0940 \u092a\u093f\u091b\u0932\u0940 \u090a\u0901\u091a\u093e\u0908 \u0915\u0947 60% \u0924\u0915 \u090a\u092a\u0930 \u091c\u093e\u0924\u0940 \u0939\u0948\u0964 (i) 5 \u0909\u091b\u093e\u0932\u094b\u0902 \u0915\u0947 \u092c\u093e\u0926 \u0917\u0947\u0902\u0926 \u0915\u093f\u0924\u0928\u0940 \u090a\u0901\u091a\u093e\u0908 \u0924\u0915 \u091c\u093e\u0924\u0940 \u0939\u0948? (ii) \u0917\u0947\u0902\u0926 \u0926\u094d\u0935\u093e\u0930\u093e \u092d\u0942\u092e\u093f \u0938\u0947 \u091b\u0920\u0940 \u092c\u093e\u0930 \u091f\u0915\u0930\u093e\u0928\u0947 \u0924\u0915 \u0924\u092f \u0915\u0940 \u0917\u0908 \u0915\u0941\u0932 \u0932\u0902\u092c\u0935\u0924 \u0926\u0942\u0930\u0940 \u0915\u093f\u0924\u0928\u0940 \u0939\u0948?<\/b><br>\nGP a=80\u00d70.6=48 r=0.6<br>\ntn=48\u00d70.6^(n-1)<br>\n(i) 5 \u0909\u091b\u093e\u0932 t5=48\u00d70.6^4=48\u00d70.1296=6.2208 \u092e\u0940<br>\n<span style=\"background:#e8f5e9;padding:3px 10px;border-radius:6px;font-weight:bold;\">\u0909\u0924\u094d\u0924\u0930 6.22 \u092e\u0940<\/span><br><br>\n(ii) \u0915\u0941\u0932 \u0926\u0942\u0930\u0940 = \u092a\u094d\u0930\u093e\u0930\u0902\u092d\u093f\u0915 \u0917\u093f\u0930\u093e\u0935\u091f 80 + 2\u00d7(\u092a\u094d\u0930\u0925\u092e 5 \u0909\u091b\u093e\u0932\u094b\u0902 \u0915\u093e \u092f\u094b\u0917 \u090a\u092a\u0930-\u0928\u0940\u091a\u0947?) \u092a\u094d\u0930\u0936\u094d\u0928 \u091b\u0920\u0940 \u092c\u093e\u0930 \u091f\u0915\u0930\u093e\u0928\u0947 \u0924\u0915:<br>\n\u0917\u093f\u0930\u0928\u093e 80 + \u090a\u092a\u0930 48 + \u0928\u0940\u091a\u0947 48 + \u090a\u092a\u0930 28.8 + \u0928\u0940\u091a\u0947 28.8 + \u090a\u092a\u0930 17.28 + \u0928\u0940\u091a\u0947 17.28 + \u090a\u092a\u0930 10.368 + \u0928\u0940\u091a\u0947 10.368 + \u090a\u092a\u0930 6.2208 + \u0928\u0940\u091a\u0947 6.2208 =?<br>\n\u0938\u0930\u0932: \u0915\u0941\u0932 =80 +2\u00d7(48+28.8+17.28+10.368+6.2208) +? \u091b\u0920\u0940 \u091f\u0915\u0930\u093e\u0928\u0947 \u0924\u0915 5 \u0909\u091b\u093e\u0932\u094b\u0902 \u0915\u093e \u0906\u0928\u093e-\u091c\u093e\u0928\u093e + \u091b\u0920\u0940 \u0909\u091b\u093e\u0932? \u092f\u0926\u093f \u091b\u0920\u0940 \u091f\u0915\u0930\u093e\u0928\u0947 \u0924\u0915 \u0924\u094b 5 \u0909\u091b\u093e\u0932\u094b\u0902 \u0915\u093e \u090a\u092a\u0930-\u0928\u0940\u091a\u0947:<br>\n\u0926\u0942\u0930\u0940 =80 +2\u00d7S5 \u091c\u0939\u093e\u0901 S5=48(1-0.6^5)\/(1-0.6)=48(1-0.07776)\/0.4=48\u00d70.92224\/0.4=110.6688<br>\n\u0915\u0941\u0932=80+2\u00d7110.6688=301.3376 \u092e\u0940<br>\n\u092f\u0926\u093f 6 \u0909\u091b\u093e\u0932\u094b\u0902 \u0924\u0915 \u090a\u092a\u0930 \u0936\u093e\u092e\u093f\u0932 \u0924\u094b +3.73248=305.07 \u092e\u0940<br>\n<span style=\"background:#e8f5e9;padding:3px 10px;border-radius:6px;font-weight:bold;\">\u0909\u0924\u094d\u0924\u0930 \u0932\u0917\u092d\u0917 301.34 \u092e\u0940 \u091f\u0915\u0930\u093e\u0928\u0947 \u0924\u0915, 305 \u092e\u0940 6\u0935\u0940\u0902 \u0909\u091b\u093e\u0932 \u0938\u0939\u093f\u0924<\/span><br>\n\n<\/div><\/div>\n\n<div id=\"q83-3\" style=\"display:none;\"><div style=\"background:#2e7d32;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udfe2 \u092d\u093e\u0917 3\/4 - Q6 Q7 \u0915\u093e\u0930\u092a\u0947\u091f<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:14px;line-height:1.9;height:560px;overflow-y:auto;\">\n\n<b>Q6. \u0905\u0928\u0941\u0915\u094d\u0930\u092e 2,2\u221a2,4,4\u221a2,... \u0915\u093e \u0915\u094c\u0928-\u0938\u093e \u092a\u0926 128 \u0939\u0948?<\/b><br>\na=2 r=\u221a2<br>\ntn=2\u00d7(\u221a2)^(n-1)=128\u2192(\u221a2)^(n-1)=64=2^6=(\u221a2)^12\u2192n-1=12\u2192n=13<br>\n<span style=\"background:#e8f5e9;padding:3px 10px;border-radius:6px;font-weight:bold;\">\u0909\u0924\u094d\u0924\u0930 13\u0935\u093e\u0901 \u092a\u0926<\/span><br><br>\n\n<b>Q7. \u091a\u093f\u0924\u094d\u0930 8.12 \u092e\u0947\u0902 \u0938\u093f\u0930\u092a\u093f\u0928\u094d\u0938\u094d\u0915\u0940 \u0935\u0930\u094d\u0917 \u0915\u093e\u0930\u092a\u0947\u091f \u0915\u0947 \u091a\u0930\u0923 0 \u0938\u0947 3 \u0924\u0915 \u092a\u094d\u0930\u0926\u0930\u094d\u0936\u093f\u0924 \u0939\u0948\u0902\u0964 \u091a\u0930\u0923 0 \u0915\u093e\u0917\u091c \u0915\u093e \u090f\u0915 \u0935\u0930\u094d\u0917\u093e\u0915\u093e\u0930 \u092a\u0943\u0937\u094d\u0920 \u0939\u0948\u0964 \u091a\u0930\u0923 1 \u0915\u0940 \u0930\u091a\u0928\u093e \u0915\u0930\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u0935\u0930\u094d\u0917 \u0915\u0940 \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u092d\u0941\u091c\u093e \u0915\u094b \u0924\u0940\u0928 \u0938\u092e\u093e\u0928 \u092d\u093e\u0917\u094b\u0902 \u092e\u0947\u0902 \u092c\u093e\u0901\u091f\u093e \u0917\u092f\u093e \u0914\u0930 \u0938\u092e\u094d\u092e\u0941\u0916 \u092d\u0941\u091c\u093e\u0913\u0902 \u0915\u0947 \u0935\u093f\u092d\u093e\u091c\u0915 \u092c\u093f\u0902\u0926\u0941\u0913\u0902 \u0915\u094b \u092e\u093f\u0932\u093e\u0928\u0947 \u092a\u0930 \u0928\u094c \u091b\u094b\u091f\u0947 \u0935\u0930\u094d\u0917 \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0939\u094b\u0924\u0947 \u0939\u0948\u0902\u0964 \u0924\u0924\u094d\u092a\u0936\u094d\u091a\u093e\u0924 \u092e\u0927\u094d\u092f \u092e\u0947\u0902 \u0938\u094d\u0925\u093f\u0924 \u0935\u0930\u094d\u0917 \u0915\u094b \u0939\u091f\u093e \u0926\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948 \u0914\u0930 8 \u091b\u094b\u091f\u0947 \u0935\u0930\u094d\u0917\u094b\u0902 \u0915\u0947 \u092e\u0927\u094d\u092f \u090f\u0915 \u0935\u0930\u094d\u0917\u093e\u0915\u093e\u0930 \u0930\u093f\u0915\u094d\u0924\u093f \u0936\u0947\u0937 \u0930\u0939 \u091c\u093e\u0924\u0940 \u0939\u0948\u0964 \u092f\u0939\u0940 \u092a\u094d\u0930\u0915\u094d\u0930\u093f\u092f\u093e \u0906\u0920 \u091b\u094b\u091f\u0947 \u091b\u093e\u092f\u093e\u0902\u0915\u093f\u0924 \u0935\u0930\u094d\u0917\u094b\u0902 \u092a\u0930 \u0926\u094b\u0939\u0930\u093e\u0908 \u091c\u093e\u0924\u0940 \u0939\u0948, \u091c\u093f\u0938\u0938\u0947 \u091a\u0930\u0923 2 \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0939\u094b\u0924\u093e \u0939\u0948 \u0914\u0930 \u0906\u0917\u0947 \u092d\u0940 \u092f\u0939\u0940 \u092a\u094d\u0930\u0915\u094d\u0930\u093f\u092f\u093e \u091c\u093e\u0930\u0940 \u0930\u0939\u0924\u0940 \u0939\u0948\u0964<\/b><br><br>\n\n(i) \u091a\u0930\u0923 0 \u0938\u0947 3 \u0924\u0915 \u0915\u093f\u0924\u0928\u0947 \u0932\u093e\u0932 \u0935\u0930\u094d\u0917 \u0939\u0948\u0902?<br>\n\u091a\u0930\u09230=1, \u091a\u0930\u09231=8, \u091a\u0930\u09232=8\u00d78=64, \u091a\u0930\u09233=64\u00d78=512<br><br>\n\n(ii) \u091a\u0930\u0923 4 \u0938\u0947 5 \u092e\u0947\u0902 \u0932\u093e\u0932 \u0935\u0930\u094d\u0917\u094b\u0902 \u0915\u0940 \u0938\u0902\u0916\u094d\u092f\u093e \u0915\u093e \u092a\u0942\u0930\u094d\u0935\u093e\u0928\u0941\u092e\u093e\u0928?<br>\n\u091a\u0930\u09234=512\u00d78=4096, \u091a\u0930\u09235=32768<br><br>\n\n(iii) n\u0935\u0947\u0902 \u091a\u0930\u0923 \u092e\u0947\u0902 \u0932\u093e\u0932 \u0935\u0930\u094d\u0917\u094b\u0902 \u0915\u0940 \u0938\u0902\u0916\u094d\u092f\u093e \u0915\u0947 \u0932\u093f\u090f \u0938\u0942\u0924\u094d\u0930? \u0935\u094d\u092f\u093e\u092a\u0915 \u0914\u0930 \u092a\u0941\u0928\u0930\u093e\u0935\u0930\u094d\u0924\u0940<br>\n\u0935\u094d\u092f\u093e\u092a\u0915 tn=8^n, \u091a\u0930\u09230 \u0938\u0947 n \u0924\u0915? \u092f\u0926\u093f \u091a\u0930\u09230 \u0915\u094b 1 \u092e\u093e\u0928\u0947\u0902 \u0924\u094b \u091a\u0930\u0923 n \u092e\u0947\u0902 8^n<br>\n\u092a\u0941\u0928\u0930\u093e\u0935\u0930\u094d\u0924\u0940 t0=1, tn=8\u00d7tn-1<br><br>\n\n(iv) \u091a\u0930\u09230 \u0915\u094d\u0937\u0947\u0924\u094d\u0930\u092b\u0932 1\u0964 \u091a\u0930\u09231,2,3 \u092e\u0947\u0902 \u0932\u093e\u0932 \u0915\u094d\u0937\u0947\u0924\u094d\u0930 \u0915\u093e \u0915\u094d\u0937\u0947\u0924\u094d\u0930\u092b\u0932? \u091a\u0930\u09234,5? n\u0935\u0947\u0902 \u091a\u0930\u0923 \u092e\u0947\u0902? \u0935\u094d\u092f\u093e\u092a\u0915 \u0914\u0930 \u092a\u0941\u0928\u0930\u093e\u0935\u0930\u094d\u0924\u0940\u0964 \u091a\u0930\u0923 \u0938\u0902\u0916\u094d\u092f\u093e n \u092c\u0922\u093c\u0928\u0947 \u0915\u0947 \u0938\u093e\u0925 \u0907\u0938 \u0915\u094d\u0937\u0947\u0924\u094d\u0930\u092b\u0932 \u092e\u0947\u0902 \u0915\u094d\u092f\u093e \u0905\u0902\u0924\u0930 \u0906\u0924\u093e \u0939\u0948?<br>\n\u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u091a\u0930\u0923 \u092e\u0947\u0902 1\/9 \u092d\u093e\u0917 \u0939\u091f\u0924\u093e, 8\/9 \u0936\u0947\u0937<br>\n\u091a\u0930\u09230=1<br>\n\u091a\u0930\u09231=8\/9\u22480.888<br>\n\u091a\u0930\u09232=(8\/9)\u00b2=64\/81\u22480.79<br>\n\u091a\u0930\u09233=(8\/9)\u00b3=512\/729\u22480.702<br>\n\u091a\u0930\u09234=(8\/9)^4\u22480.624, \u091a\u0930\u09235=(8\/9)^5\u22480.554<br>\n\u0935\u094d\u092f\u093e\u092a\u0915 sn=(8\/9)^n<br>\n\u092a\u0941\u0928\u0930\u093e\u0935\u0930\u094d\u0924\u0940 s0=1, sn=8\/9\u00d7sn-1<br>\nn \u092c\u0922\u093c\u0928\u0947 \u092a\u0930 \u0915\u094d\u0937\u0947\u0924\u094d\u0930\u092b\u0932 \u0918\u091f\u0924\u093e \u0939\u0948 \u0914\u0930 0 \u0915\u0940 \u0913\u0930 \u091c\u093e\u0924\u093e \u0939\u0948, \u091c\u092c\u0915\u093f \u0935\u0930\u094d\u0917\u094b\u0902 \u0915\u0940 \u0938\u0902\u0916\u094d\u092f\u093e \u0924\u0947\u091c\u0940 \u0938\u0947 \u092c\u0922\u093c\u0924\u0940 \u0939\u0948\u0964<br>\n<span style=\"background:#e8f5e9;padding:3px 10px;border-radius:6px;font-weight:bold;\">\u0909\u0924\u094d\u0924\u0930 GP r8 \u0938\u0902\u0916\u094d\u092f\u093e, r8\/9 \u0915\u094d\u0937\u0947\u0924\u094d\u0930\u092b\u0932<\/span><br>\n\n<\/div><\/div>\n\n<div id=\"q83-4\" style=\"display:none;\"><div style=\"background:#6a1b9a;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udfe3 \u092d\u093e\u0917 4\/4 - FINAL<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:14px;line-height:2;height:560px;overflow-y:auto;\">\n\n<div style=\"background:#fff9c4;border:3px solid #fbc02d;padding:10px;border-radius:10px;text-align:center;font-weight:bold;\">\n\u2705 \u092a\u094d\u0930\u0936\u094d\u0928\u093e\u0935\u0932\u0940 8.3 Q1-Q7 \u0936\u0941\u0926\u094d\u0927 \u0916\u0924\u094d\u092e<br><br>\nQ1 r2 8\u0935\u093e\u0901 192 a1.5 t12=3072<br>\nQ2 5,25... tn5^n t10=9765625<br>\nQ3 t1=2 tn+1=3tn-2 \u21922,4,10,28,82,244,730 7\u0935\u093e\u0901 \u092a\u0926<br>\nQ4 2,6,18... 8\u0935\u093e\u0901 4374 tn2\u00d73^(n-1) \u092a\u0941\u0928\u0930\u093e\u0935\u0930\u094d\u0924\u0940 t1=2 tn=3tn-1<br>\nQ5 80\u092e\u0940 60% GP a48 r0.6 t5=6.22\u092e\u0940 \u0915\u0941\u0932 6 \u091f\u0915\u0930\u093e\u0928\u0947 \u0924\u0915 ~301.34\u092e\u0940<br>\nQ6 2,2\u221a2,4... r\u221a2 13\u0935\u093e\u0901 128<br>\nQ7 \u0938\u093f\u0930\u092a\u093f\u0928\u094d\u0938\u094d\u0915\u0940 \u0915\u093e\u0930\u092a\u0947\u091f \u0932\u093e\u0932 \u0935\u0930\u094d\u0917 1,8,64,512,4096,32768... tn8^n \u0915\u094d\u0937\u0947\u0924\u094d\u0930\u092b\u0932 1,8\/9,(8\/9)\u00b2... sn(8\/9)^n \u0918\u091f\u0915\u0930 0\n<\/div>\n\n<\/div><\/div>\n<\/div>\n\n<div style=\"background:#0a5c4b;padding:12px;display:flex;justify-content:space-between;align-items:center;\">\n<button type=\"button\" id=\"q83p\" style=\"background:#ffcc00;color:#000;border:3px solid #fff;border-radius:30px;padding:10px 22px;font-weight:bold;cursor:pointer;min-width:100px;\">&larr; \u092a\u0940\u091b\u0947<\/button>\n<span id=\"q83i\" style=\"background:#fff;color:#0a5c4b;padding:8px 20px;border-radius:20px;font-weight:bold;\">1 \/ 4<\/span>\n<button type=\"button\" id=\"q83n\" style=\"background:#00e676;color:#000;border:3px solid #fff;border-radius:30px;padding:10px 22px;font-weight:bold;cursor:pointer;min-width:100px;\">\u0906\u0917\u0947 &rarr;<\/button>\n<\/div>\n<\/div>\n\n<script>\n(function(){\nvar c=1,t=4;\nfunction s(n){\nfor(var i=1;i<=t;i++){\nvar el=document.getElementById('q83-'+i);\nif(el)el.style.display=(i===n?'block':'none');\n}\ndocument.getElementById('q83i').textContent=n+' \/ '+t;\nc=n;\n}\ndocument.getElementById('q83n').onclick=function(){var n=c+1;if(n>t)n=1;s(n);};\ndocument.getElementById('q83p').onclick=function(){var n=c-1;if(n<1)n=t;s(n);};\n})();\n<\/script><\/div><\/div><\/div><\/div><div id=\"pg-991-7\"  class=\"panel-grid panel-no-style\" ><div id=\"pgc-991-7-0\"  class=\"panel-grid-cell\" ><div id=\"panel-991-7-0-0\" class=\"widget_text so-panel widget widget_custom_html panel-first-child panel-last-child\" data-index=\"7\" ><div class=\"textwidget custom-html-widget\"><div id=\"QF-ALL\" style=\"max-width:860px;margin:12px auto;border:4px solid #0f5e4b;border-radius:14px;background:#e6f4f1;font-family:'Noto Sans Devanagari',Arial,sans-serif;overflow:hidden;\">\n<div style=\"background:#0a5c4b;padding:10px;text-align:center;\"><span style=\"background:#fff;color:#0a3d33;padding:7px 20px;border-radius:20px;font-weight:bold;font-size:13px;\">\u0905\u0902\u0924\u093f\u092e \u092a\u094d\u0930\u0936\u094d\u0928\u093e\u0935\u0932\u0940 Q1-15 - \u092a\u0947\u091c 195-196 \u0936\u0941\u0926\u094d\u0927<\/span><\/div>\n\n<div style=\"padding:10px 12px;\">\n<div id=\"qf-1\" style=\"display:block;\"><div style=\"background:#d32f2f;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udd34 \u092d\u093e\u0917 1\/5 - Q1-Q5<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:14px;line-height:1.9;height:560px;overflow-y:auto;\">\n\n<b>Q1. \u0909\u0938 \u0938\u092e\u093e\u0928\u094d\u0924\u0930 \u0936\u094d\u0930\u0947\u0922\u093c\u0940 \u0915\u093e 31\u0935\u093e\u0901 \u092a\u0926 \u091c\u094d\u091e\u093e\u0924 \u0915\u0940\u091c\u093f\u090f, \u091c\u093f\u0938\u0915\u093e 11\u0935\u093e\u0901 \u092a\u0926 38 \u0939\u0948 \u0914\u0930 16\u0935\u093e\u0901 \u092a\u0926 73 \u0939\u0948\u0964<\/b><br>\na+10d=38, a+15d=73\u21925d=35\u2192d=7, a=38-70=-32<br>\nt31=a+30d=-32+210=178<br>\n<span style=\"background:#e8f5e9;padding:3px 10px;border-radius:6px;font-weight:bold;\">\u0909\u0924\u094d\u0924\u0930 178<\/span><br><br>\n\n<b>Q2. \u0909\u0938 \u0938\u092e\u093e\u0928\u094d\u0924\u0930 \u0936\u094d\u0930\u0947\u0922\u093c\u0940 \u0915\u093e \u0928\u093f\u0930\u094d\u0927\u093e\u0930\u0923 \u0915\u0940\u091c\u093f\u090f, \u091c\u093f\u0938\u0915\u093e \u0924\u0940\u0938\u0930\u093e \u092a\u0926 16 \u0939\u0948 \u0914\u0930 \u091c\u093f\u0938\u0915\u093e 7\u0935\u093e\u0901 \u092a\u0926, 5\u0935\u0947\u0902 \u092a\u0926 \u0938\u0947 12 \u0905\u0927\u093f\u0915 \u0939\u0948\u0964<\/b><br>\na+2d=16, (a+6d)=(a+4d)+12\u21922d=12\u2192d=6, a=16-12=4<br>\nAP 4,10,16,22...<br>\n<span style=\"background:#e8f5e9;padding:3px 10px;border-radius:6px;font-weight:bold;\">\u0909\u0924\u094d\u0924\u0930 4,10,16,22...<\/span><br><br>\n\n<b>Q3. \u0915\u093f\u0924\u0928\u0940 3-\u0905\u0902\u0915\u0940\u092f \u0938\u0902\u0916\u094d\u092f\u093e\u090f\u0901 7 \u0938\u0947 \u0935\u093f\u092d\u093e\u091c\u094d\u092f \u0939\u0948\u0902?<\/b><br>\n\u0938\u092c\u0938\u0947 \u091b\u094b\u091f\u0940 3-\u0905\u0902\u0915\u0940\u092f 7 \u0938\u0947 \u0935\u093f\u092d\u093e\u091c\u094d\u092f 105, \u0938\u092c\u0938\u0947 \u092c\u0921\u093c\u0940 994<br>\nAP 105,112...994, d7, 994=105+(n-1)7\u2192889=7(n-1)\u2192n-1=127\u2192n=128<br>\n<span style=\"background:#e8f5e9;padding:3px 10px;border-radius:6px;font-weight:bold;\">\u0909\u0924\u094d\u0924\u0930 128 \u0938\u0902\u0916\u094d\u092f\u093e\u090f\u0901<\/span><br><br>\n\n<b>Q4. 4 \u0915\u0947 \u0915\u093f\u0924\u0928\u0947 \u0917\u0941\u0923\u091c 10 \u0914\u0930 250 \u0915\u0947 \u092c\u0940\u091a \u0906\u0924\u0947 \u0939\u0948\u0902?<\/b><br>\n10-250 \u092c\u0940\u091a 4 \u0915\u0947 \u0917\u0941\u0923\u091c: \u092a\u0939\u0932\u093e 12, \u0905\u0902\u0924\u093f\u092e 248<br>\nAP 12,16...248 d4, 248=12+(n-1)4\u2192236=4(n-1)\u2192n-1=59\u2192n=60<br>\n<span style=\"background:#e8f5e9;padding:3px 10px;border-radius:6px;font-weight:bold;\">\u0909\u0924\u094d\u0924\u0930 60 \u0917\u0941\u0923\u091c<\/span><br><br>\n\n<b>Q5. \u0909\u0938 \u0917\u0941\u0923\u094b\u0924\u094d\u0924\u0930 \u0936\u094d\u0930\u0947\u0922\u093c\u0940 \u0915\u0940 \u0930\u091a\u0928\u093e \u0915\u0940\u091c\u093f\u090f, \u091c\u093f\u0938\u0915\u0947 \u092a\u094d\u0930\u093e\u0930\u0902\u092d\u093f\u0915 \u0926\u094b \u092a\u0926\u094b\u0902 \u0915\u093e \u092f\u094b\u0917\u092b\u0932 -4 \u0939\u0948 \u0914\u0930 \u091c\u093f\u0938\u0915\u093e \u092a\u093e\u0901\u091a\u0935\u093e \u092a\u0926, \u0924\u0940\u0938\u0930\u0947 \u092a\u0926 \u0915\u093e 4 \u0917\u0941\u0928\u093e \u0939\u0948\u0964<\/b><br>\na+ar=-4\u2192a(1+r)=-4, t5=4\u00d7t3\u2192ar^4=4ar^2\u2192r\u00b2=4\u2192r=\u00b12<br>\n\u092f\u0926\u093f r=2: a(3)=-4\u2192a=-4\/3 GP -4\/3,-8\/3,-16\/3...<br>\n\u092f\u0926\u093f r=-2: a(1-2)=-4\u2192-a=-4\u2192a=4 GP 4,-8,16,-32...<br>\n<span style=\"background:#e8f5e9;padding:3px 10px;border-radius:6px;font-weight:bold;\">\u0909\u0924\u094d\u0924\u0930 r=2 a=-4\/3 \u092f\u093e r=-2 a=4 \u0926\u094b GP<\/span><br>\n\n<\/div><\/div>\n\n<div id=\"qf-2\" style=\"display:none;\"><div style=\"background:#1565c0;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udd35 \u092d\u093e\u0917 2\/5 - Q6-Q8<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:14px;line-height:1.9;height:560px;overflow-y:auto;\">\n\n<b>Q6. 100 \u0915\u094b \u0915\u094d\u0930\u092e\u093e\u0917\u0924 \u092a\u094d\u0930\u093e\u0915\u0943\u0924 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0947 \u092f\u094b\u0917 \u0915\u0947 \u0930\u0942\u092a \u092e\u0947\u0902 \u0932\u093f\u0916\u0928\u0947 \u0915\u0947 \u0938\u092d\u0940 \u0938\u0902\u092d\u0935 \u0935\u093f\u0927\u093f\u092f\u093e\u0901 \u091c\u094d\u091e\u093e\u0924 \u0915\u0940\u091c\u093f\u090f\u0964<\/b><br>\n\u0915\u094d\u0930\u092e\u093e\u0917\u0924 \u092f\u094b\u0917: k \u0938\u0947 k+m-1 \u0924\u0915 =m\/2[2k+m-1]=100<br>\n\u0935\u093f\u0927\u093f\u092f\u093e\u0901:<br>\n9+10+11+12+13+14+15+16=100 (8 \u092a\u0926 9-16)<br>\n18+19+20+21+22=100 (5 \u092a\u0926)<br>\n100 \u0905\u0915\u0947\u0932\u0947 \u092d\u0940? \u092a\u0930 \u0915\u094d\u0930\u092e\u093e\u0917\u0924 2 \u092f\u093e \u0905\u0927\u093f\u0915:<br>\n\u0939\u0932 m(m+2k-1)=200, m \u0915\u0947 \u0917\u0941\u0923\u0928\u0916\u0902\u0921 \u091c\u094b \u0935\u093f\u0937\u092e\/\u0938\u092e<br>\n\u0938\u092d\u0940: 100, 18-22, 9-16<br>\n<span style=\"background:#e8f5e9;padding:3px 10px;border-radius:6px;font-weight:bold;\">\u0909\u0924\u094d\u0924\u0930 9 \u0938\u0947 16 \u0924\u0915 \u0914\u0930 18 \u0938\u0947 22 \u0924\u0915 (\u0914\u0930 100 \u0938\u094d\u0935\u092f\u0902)<\/span><br><br>\n\n<b>Q7. \u0915\u093f\u0938\u0940 \u0915\u0932\u094d\u091a\u0930 \u092e\u0947\u0902 \u091c\u0940\u0935\u093e\u0923\u0941\u0913\u0902 \u0915\u0940 \u0938\u0902\u0916\u094d\u092f\u093e \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u0918\u0902\u091f\u0947 \u092e\u0947\u0902 \u0926\u094b\u0917\u0941\u0928\u0940 \u0939\u094b \u091c\u093e\u0924\u0940 \u0939\u0948\u0964 \u092f\u0926\u093f \u092a\u094d\u0930\u093e\u0930\u0902\u092d \u092e\u0947\u0902 \u0915\u0932\u094d\u091a\u0930 \u092e\u0947\u0902 30 \u091c\u0940\u0935\u093e\u0923\u0941 \u0935\u093f\u0926\u094d\u092f\u092e\u093e\u0928 \u0925\u0947, \u0924\u094b \u0926\u0942\u0938\u0930\u0947 \u0918\u0902\u091f\u0947 \u0915\u0947 \u0905\u0902\u0924 \u092e\u0947\u0902, \u091a\u094c\u0925\u0947 \u0918\u0902\u091f\u0947 \u0915\u0947 \u0905\u0902\u0924 \u092e\u0947\u0902 \u0914\u0930 n\u0935\u0947\u0902 \u0918\u0902\u091f\u0947 \u0915\u0947 \u0905\u0902\u0924 \u092e\u0947\u0902 \u0915\u093f\u0924\u0928\u0947 \u091c\u0940\u0935\u093e\u0923\u0941 \u0935\u093f\u0926\u094d\u092f\u092e\u093e\u0928 \u0939\u094b\u0902\u0917\u0947?<\/b><br>\nGP a30 r2<br>\n1 \u0918\u0902\u091f\u0947 \u092c\u093e\u0926 60, 2 \u0918\u0902\u091f\u0947 120, 4 \u0918\u0902\u091f\u0947 480, n \u0918\u0902\u091f\u0947 30\u00d72^n<br>\n\u092f\u0926\u093f \u092a\u094d\u0930\u093e\u0930\u0902\u092d 0 \u0918\u0902\u091f\u0947 30 \u0924\u094b t_n=30\u00d72^n<br>\n<span style=\"background:#e8f5e9;padding:3px 10px;border-radius:6px;font-weight:bold;\">\u0909\u0924\u094d\u0924\u0930 2 \u0918\u0902\u091f\u0947 120, 4 \u0918\u0902\u091f\u0947 480, n \u0918\u0902\u091f\u0947 30\u00d72^n<\/span><br><br>\n\n<b>Q8. \u0915\u093f\u0938\u0940 \u0938\u092e\u093e\u0928\u094d\u0924\u0930 \u0936\u094d\u0930\u0947\u0922\u093c\u0940 \u0915\u0947 \u091a\u094c\u0925\u0947 \u0914\u0930 \u0906\u0920\u0935\u0947\u0902 \u092a\u0926\u094b\u0902 \u0915\u093e \u092f\u094b\u0917\u092b\u0932 24 \u0939\u0948 \u0914\u0930 \u091b\u0920\u0947 \u0914\u0930 \u0926\u0938\u0935\u0947\u0902 \u092a\u0926\u094b\u0902 \u0915\u093e \u092f\u094b\u0917\u092b\u0932 44 \u0939\u0948\u0964 \u0938\u092e\u093e\u0928\u094d\u0924\u0930 \u0936\u094d\u0930\u0947\u0922\u093c\u0940 \u0915\u0947 \u092a\u0939\u0932\u0947 \u0924\u0940\u0928 \u092a\u0926 \u091c\u094d\u091e\u093e\u0924 \u0915\u0940\u091c\u093f\u090f\u0964<\/b><br>\nt4+t8= (a+3d)+(a+7d)=2a+10d=24\u2192a+5d=12<br>\nt6+t10= (a+5d)+(a+9d)=2a+14d=44\u2192a+7d=22<br>\n2d=10\u2192d=5, a=12-25=-13<br>\n\u092a\u0939\u0932\u0947 \u0924\u0940\u0928 -13,-8,-3<br>\n<span style=\"background:#e8f5e9;padding:3px 10px;border-radius:6px;font-weight:bold;\">\u0909\u0924\u094d\u0924\u0930 -13,-8,-3 AP<\/span><br>\n\n<\/div><\/div>\n\n<div id=\"qf-3\" style=\"display:none;\"><div style=\"background:#2e7d32;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udfe2 \u092d\u093e\u0917 3\/5 - Q9-Q12<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:14px;line-height:1.9;height:560px;overflow-y:auto;\">\n\n<b>Q9. n \u0915\u093e \u0935\u0939 \u0928\u094d\u092f\u0942\u0928\u0924\u092e \u092e\u093e\u0928 \u091c\u094d\u091e\u093e\u0924 \u0915\u0940\u091c\u093f\u090f, \u091c\u093f\u0938\u0938\u0947 \u092a\u094d\u0930\u0925\u092e n \u092a\u094d\u0930\u093e\u0915\u0943\u0924 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u093e \u092f\u094b\u0917\u092b\u0932 1,000 \u0938\u0947 \u0905\u0927\u093f\u0915 \u0939\u094b\u0964<\/b><br>\nn(n+1)\/2>1000\u2192n\u00b2+n-2000>0<br>\nn\u224844: 44\u00d745\/2=990<1000, 45\u00d746\/2=1035>1000<br>\n<span style=\"background:#e8f5e9;padding:3px 10px;border-radius:6px;font-weight:bold;\">\u0909\u0924\u094d\u0924\u0930 n=45<\/span><br><br>\n\n<b>Q10. \u0917\u0941\u0923\u094b\u0924\u094d\u0924\u0930 \u0936\u094d\u0930\u0947\u0922\u093c\u0940 2,8,32,... \u0915\u093e \u0915\u094c\u0928-\u0938\u093e \u092a\u0926 131072 \u0939\u0948? n\u0935\u0947\u0902 \u092a\u0926 \u0915\u0947 \u0932\u093f\u090f \u0935\u094d\u092f\u093e\u092a\u0915 \u0938\u0942\u0924\u094d\u0930 \u0914\u0930 \u092a\u0941\u0928\u0930\u093e\u0935\u0930\u094d\u0924\u0940 \u0938\u0942\u0924\u094d\u0930 \u092d\u0940 \u0932\u093f\u0916\u093f\u090f\u0964<\/b><br>\na2 r4 tn=2\u00d74^(n-1)=2^(2n-1)=131072=2^17\u21922n-1=17\u2192n=9<br>\n\u0935\u094d\u092f\u093e\u092a\u0915 tn=2\u00d74^(n-1), \u092a\u0941\u0928\u0930\u093e\u0935\u0930\u094d\u0924\u0940 t1=2 tn=4tn-1<br>\n<span style=\"background:#e8f5e9;padding:3px 10px;border-radius:6px;font-weight:bold;\">\u0909\u0924\u094d\u0924\u0930 9\u0935\u093e\u0901 \u092a\u0926<\/span><br><br>\n\n<b>Q11. \u090f\u0915 \u0917\u0941\u0923\u094b\u0924\u094d\u0924\u0930 \u0936\u094d\u0930\u0947\u0922\u093c\u0940 \u0915\u0947 \u092a\u094d\u0930\u0925\u092e \u0924\u0940\u0928 \u092a\u0926\u094b\u0902 \u0915\u093e \u092f\u094b\u0917\u092b\u0932 13\/12 \u0939\u0948 \u0914\u0930 \u0909\u0928\u0915\u093e \u0917\u0941\u0923\u0928\u092b\u0932 -1 \u0939\u0948\u0964 \u0936\u094d\u0930\u0947\u0922\u093c\u0940 \u0915\u093e \u0938\u093e\u0930\u094d\u0935 \u0905\u0928\u0941\u092a\u093e\u0924 \u0914\u0930 \u092a\u0926 \u091c\u094d\u091e\u093e\u0924 \u0915\u0940\u091c\u093f\u090f\u0964<\/b><br>\n\u092e\u093e\u0928\u093e a\/r, a, ar \u2192 \u0917\u0941\u0923\u0928\u092b\u0932 a\u00b3=-1\u2192a=-1<br>\n\u092f\u094b\u0917 -1\/r -1 -r =13\/12\u2192-(1\/r+r+1)=13\/12\u21921\/r+r= -25\/12<br>\nr+1\/r=-25\/12\u219212r\u00b2+25r+12=0\u2192(3r+4)(4r+3)=0\u2192r=-4\/3 \u092f\u093e -3\/4<br>\n\u092a\u0926: r=-4\/3 \u2192 3\/4,-1,4\/3 ; r=-3\/4 \u2192 4\/3,-1,3\/4<br>\n<span style=\"background:#e8f5e9;padding:3px 10px;border-radius:6px;font-weight:bold;\">\u0909\u0924\u094d\u0924\u0930 r=-4\/3 \u092f\u093e -3\/4 \u092a\u0926 3\/4,-1,4\/3<\/span><br><br>\n\n<b>Q12. \u092f\u0926\u093f \u0915\u093f\u0938\u0940 \u0917\u0941\u0923\u094b\u0924\u094d\u0924\u0930 \u0936\u094d\u0930\u0947\u0922\u093c\u0940 \u0915\u093e \u091a\u094c\u0925\u093e, \u0926\u0938\u0935\u093e\u0901 \u0914\u0930 \u0938\u094b\u0932\u0939\u0935\u093e\u0901 \u092a\u0926 \u0915\u094d\u0930\u092e\u0936\u0903 x,y,z \u0939\u0948\u0964 \u0938\u093f\u0926\u094d\u0927 \u0915\u0940\u091c\u093f\u090f \u0915\u093f x,y,z \u0917\u0941\u0923\u094b\u0924\u094d\u0924\u0930 \u0936\u094d\u0930\u0947\u0922\u093c\u0940 \u092e\u0947\u0902 \u0939\u0948\u0902\u0964<\/b><br>\nt4=ar\u00b3=x, t10=ar^9=y, t16=ar^15=z<br>\ny\u00b2=(ar^9)\u00b2=a\u00b2r^18, x\u00d7z=ar\u00b3\u00d7ar^15=a\u00b2r^18\u2192y\u00b2=xz\u2192x,y,z GP<br>\n<span style=\"background:#e8f5e9;padding:3px 10px;border-radius:6px;font-weight:bold;\">\u0938\u093f\u0926\u094d\u0927<\/span><br>\n\n<\/div><\/div>\n\n<div id=\"qf-4\" style=\"display:none;\"><div style=\"background:#6a1b9a;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udfe3 \u092d\u093e\u0917 4\/5 - Q13-Q15<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:14px;line-height:1.9;height:560px;overflow-y:auto;\">\n\n<b>Q13. \u090f\u0915 \u0917\u0941\u0923\u094b\u0924\u094d\u0924\u0930 \u0936\u094d\u0930\u0947\u0922\u093c\u0940 \u0915\u0947 \u092a\u094d\u0930\u0925\u092e \u0924\u0940\u0928 \u092a\u0926\u094b\u0902 \u0915\u093e \u092f\u094b\u0917\u092b\u0932 26 \u0939\u0948 \u0914\u0930 \u0909\u0928 \u092a\u0926\u094b\u0902 \u0915\u0947 \u0935\u0930\u094d\u0917\u094b\u0902 \u0915\u093e \u092f\u094b\u0917\u092b\u0932 364 \u0939\u0948\u0964 GP \u0915\u0947 \u092a\u0926\u094b\u0902 \u0915\u094b \u091c\u094d\u091e\u093e\u0924 \u0915\u0940\u091c\u093f\u090f\u0964<\/b><br>\na+ar+ar\u00b2=26\u2192a(1+r+r\u00b2)=26<br>\na\u00b2+a\u00b2r\u00b2+a\u00b2r^4=364\u2192a\u00b2(1+r\u00b2+r^4)=364<br>\n1+r\u00b2+r^4=(1+r+r\u00b2)(1-r+r\u00b2)? \u0928\u0939\u0940\u0902, (r\u00b2+r+1)(r\u00b2-r+1)=r^4+r\u00b2+1<br>\n\u0924\u094b a\u00b2(r\u00b2+r+1)(r\u00b2-r+1)=364<br>\n\u092e\u093e\u0928\u093e S=26\u2192a=26\/(1+r+r\u00b2)<br>\n\u2192676(r\u00b2-r+1)\/(1+r+r\u00b2)=364\u2192676r\u00b2-676r+676=364+364r+364r\u00b2\u2192312r\u00b2-1040r+312=0\u21923r\u00b2-10r+3=0\u2192(3r-1)(r-3)=0\u2192r=3 \u092f\u093e 1\/3<br>\nr3\u2192a=2 \u2192 2,6,18 ; r1\/3\u2192a=18 \u219218,6,2<br>\n<span style=\"background:#e8f5e9;padding:3px 10px;border-radius:6px;font-weight:bold;\">\u0909\u0924\u094d\u0924\u0930 2,6,18 \u092f\u093e 18,6,2<\/span><br><br>\n\n<b>Q14. \u092e\u093e\u0928 \u0932\u0940\u091c\u093f\u090f P1=1, P2=2 \u0939\u0948 \u0914\u0930 n>2 \u0915\u0947 \u0932\u093f\u090f Pn=P1+P2+...+Pn-1+1 \u0939\u0948\u0964 P1,P2,...,P8 \u0915\u0947 \u092e\u093e\u0928 \u091c\u094d\u091e\u093e\u0924 \u0915\u0940\u091c\u093f\u090f\u0964 \u0915\u094d\u092f\u093e \u0906\u092a Pn \u0915\u0947 \u0932\u093f\u090f \u0905\u0927\u093f\u0915 \u0938\u0930\u0932 \u092a\u0941\u0928\u0930\u093e\u0935\u0930\u094d\u0924\u0940 \u0938\u0942\u0924\u094d\u0930 \u091c\u094d\u091e\u093e\u0924 \u0915\u0930 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902? \u0915\u094d\u092f\u093e \u0906\u092a \u090f\u0915 \u0935\u094d\u092f\u093e\u092a\u0915 \u0938\u0942\u0924\u094d\u0930 \u092c\u0924\u093e \u0938\u0915\u0924\u0947 \u0939\u0948\u0902?<\/b><br>\nP1=1<br>\nP2=2<br>\nP3=P1+P2+1=4<br>\nP4=1+2+4+1=8<br>\nP5=1+2+4+8+1=16<br>\nP6=32, P7=64, P8=128<br>\n\u0938\u0930\u0932 \u092a\u0941\u0928\u0930\u093e\u0935\u0930\u094d\u0924\u0940: Pn=2Pn-1 (n>2, P2=2) \u0915\u094d\u092f\u094b\u0902\u0915\u093f Pn = (Pn-1-1)+Pn-1+1? \u091c\u093e\u0901\u091a: Pn-1 =P1..Pn-2+1\u2192 Pn=Pn-1+Pn-1=2Pn-1<br>\n\u0935\u094d\u092f\u093e\u092a\u0915 Pn=2^(n-1) (P1=1=2^0)<br>\n<span style=\"background:#e8f5e9;padding:3px 10px;border-radius:6px;font-weight:bold;\">\u0909\u0924\u094d\u0924\u0930 1,2,4,8,16,32,64,128 Pn=2^(n-1)<\/span><br><br>\n\n<b>Q15. \u092e\u093e\u0928 \u0932\u0940\u091c\u093f\u090f W1=1, W2=2 \u0914\u0930 n>2 \u0915\u0947 \u0932\u093f\u090f Wn=W1+W2+...+Wn-2+2 \u0939\u0948\u0964 W1,W2,...,W8 \u0915\u0947 \u092e\u093e\u0928 \u091c\u094d\u091e\u093e\u0924 \u0915\u0940\u091c\u093f\u090f\u0964 \u0915\u094d\u092f\u093e \u0906\u092a \u0907\u0938 \u0905\u0928\u0941\u0915\u094d\u0930\u092e \u0915\u094b \u092a\u0939\u091a\u093e\u0928\u0924\u0947 \u0939\u0948\u0902?<\/b><br>\nW1=1<br>\nW2=2<br>\nW3=W1+2=3<br>\nW4=W1+W2+2=5<br>\nW5=W1+W2+W3+2=8<br>\nW6=1+2+3+5+2=13<br>\nW7=1+2+3+5+8+2=21<br>\nW8=1+2+3+5+8+13+2=34<br>\n1,2,3,5,8,13,21,34 \u2192 \u092b\u093f\u092c\u094b\u0928\u093e\u091a\u0940 \u091c\u0948\u0938\u093e (\u0935\u093f\u0930\u0939\u093e\u0902\u0915-\u092b\u093f\u092c\u094b\u0928\u093e\u091a\u0940) W1=1,W2=2, Wn=Wn-1+Wn-2<br>\n\u0935\u094d\u092f\u093e\u092a\u0915 \u0938\u0930\u0932 \u092a\u0941\u0928\u0930\u093e\u0935\u0930\u094d\u0924\u0940 Wn=Wn-1+Wn-2<br>\n<span style=\"background:#e8f5e9;padding:3px 10px;border-radius:6px;font-weight:bold;\">\u0909\u0924\u094d\u0924\u0930 1,2,3,5,8,13,21,34 \u092b\u093f\u092c\u094b\u0928\u093e\u091a\u0940<\/span><br>\n\n<\/div><\/div>\n\n<div id=\"qf-5\" style=\"display:none;\"><div style=\"background:#ff6f00;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udfe0 \u092d\u093e\u0917 5\/5 - FINAL \u0938\u093e\u0930<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:14px;line-height:2;height:560px;overflow-y:auto;\">\n\n<div style=\"background:#fff9c4;border:3px solid #fbc02d;padding:10px;border-radius:10px;text-align:center;font-weight:bold;\">\n\u2705 \u0905\u0902\u0924\u093f\u092e \u092a\u094d\u0930\u0936\u094d\u0928\u093e\u0935\u0932\u0940 Q1-Q15 FINAL \u0936\u0941\u0926\u094d\u0927<br><br>\nQ1 11\u0935\u093e\u090138 16\u0935\u093e\u090173 d7 a-32 t31=178<br>\nQ2 \u0924\u0940\u0938\u0930\u093e16 7\u0935\u093e\u0901=5\u0935\u093e\u0901+12 d6 a4 AP4,10,16<br>\nQ3 3-\u0905\u0902\u0915\u0940\u092f 7 \u0938\u0947 128 \u0938\u0902\u0916\u094d\u092f\u093e 105-994<br>\nQ4 4 \u0915\u0947 \u0917\u0941\u0923\u091c 10-250 \u092c\u0940\u091a 60 \u0938\u0902\u0916\u094d\u092f\u093e 12-248<br>\nQ5 a(1+r)=-4 r\u00b2=4 r\u00b12 GP-4\/3,-8\/3... \u0914\u0930 4,-8,16...<br>\nQ6 100 \u0915\u094d\u0930\u092e\u093e\u0917\u0924 9-16 (8 \u092a\u0926) 18-22 (5 \u092a\u0926)<br>\nQ7 \u091c\u0940\u0935\u093e\u0923\u0941 30\u00d72^n 2\u0918\u0902\u091f\u0947120 4\u0918\u0902\u091f\u0947480<br>\nQ8 t4+t8=24 t6+t10=44 d5 a-13 -13,-8,-3<br>\nQ9 n(n+1)\/2>1000 n=45 1035<br>\nQ10 2,8,32 r4 9\u0935\u093e\u0901131072 tn2\u00d74^(n-1) \u092a\u0941\u0928\u0930\u093e\u0935\u0930\u094d\u0924\u0940 t1=2 tn=4tn-1<br>\nQ11 a\/r,a,ar \u0917\u0941\u0923\u0928-1 a-1 r-4\/3,-3\/4 \u092a\u09263\/4,-1,4\/3<br>\nQ12 t4=ar\u00b3=x t10=ar^9=y t16=ar^15=z y\u00b2=xz GP \u0938\u093f\u0926\u094d\u0927<br>\nQ13 a(1+r+r\u00b2)=26 a\u00b2(1+r\u00b2+r^4)=364 r3 \u092f\u093e1\/3 \u092a\u09262,6,18<br>\nQ14 P1..P8 1,2,4,8,16,32,64,128 \u0938\u0930\u0932 Pn=2Pn-1 \u0935\u094d\u092f\u093e\u092a\u09152^(n-1)<br>\nQ15 W1..W8 1,2,3,5,8,13,21,34 \u092b\u093f\u092c\u094b\u0928\u093e\u091a\u0940 Wn=Wn-1+Wn-2\n<\/div>\n\n<\/div><\/div>\n<\/div>\n\n<div style=\"background:#0a5c4b;padding:12px;display:flex;justify-content:space-between;align-items:center;\">\n<button type=\"button\" id=\"qfp\" style=\"background:#ffcc00;color:#000;border:3px solid #fff;border-radius:30px;padding:10px 22px;font-weight:bold;cursor:pointer;min-width:100px;\">&larr; \u092a\u0940\u091b\u0947<\/button>\n<span id=\"qfi\" style=\"background:#fff;color:#0a5c4b;padding:8px 20px;border-radius:20px;font-weight:bold;\">1 \/ 5<\/span>\n<button type=\"button\" id=\"qfn\" style=\"background:#00e676;color:#000;border:3px solid #fff;border-radius:30px;padding:10px 22px;font-weight:bold;cursor:pointer;min-width:100px;\">\u0906\u0917\u0947 &rarr;<\/button>\n<\/div>\n<\/div>\n\n<script>\n(function(){\nvar c=1,t=5;\nfunction s(n){\nfor(var i=1;i<=t;i++){\nvar el=document.getElementById('qf-'+i);\nif(el)el.style.display=(i===n?'block':'none');\n}\ndocument.getElementById('qfi').textContent=n+' \/ '+t;\nc=n;\n}\ndocument.getElementById('qfn').onclick=function(){var n=c+1;if(n>t)n=1;s(n);};\ndocument.getElementById('qfp').onclick=function(){var n=c-1;if(n<1)n=t;s(n);};\n})();\n<\/script><\/div><\/div><\/div><\/div><div id=\"pg-991-8\"  class=\"panel-grid panel-no-style\" ><div id=\"pgc-991-8-0\"  class=\"panel-grid-cell\" ><div id=\"panel-991-8-0-0\" class=\"widget_text so-panel widget widget_custom_html panel-first-child panel-last-child\" data-index=\"8\" ><div class=\"textwidget custom-html-widget\"><div id=\"AG-8\" style=\"max-width:860px;margin:12px auto;border:4px solid #0f5e4b;border-radius:14px;background:#e6f4f1;font-family:'Noto Sans Devanagari',Arial,sans-serif;overflow:hidden;\">\n<div style=\"background:#0a5c4b;padding:10px;text-align:center;\"><span style=\"background:#fff;color:#0a3d33;padding:7px 20px;border-radius:20px;font-weight:bold;font-size:13px;\">\u0905\u0927\u094d\u092f\u093e\u092f 8 \u0938\u0947 \u0905\u0917\u094d\u0928\u093f\u0935\u0940\u0930 - 20 MCQ \u092a\u0915\u094d\u0915\u093e<\/span><\/div>\n\n<div style=\"padding:10px 12px;\">\n<div id=\"ag8-1\" style=\"display:block;\"><div style=\"background:#d32f2f;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udd34 \u092d\u093e\u0917 1\/4 - Q1-Q6<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:14px;line-height:1.9;height:560px;overflow-y:auto;\">\n\n<b>1. \u0905\u0928\u0941\u0915\u094d\u0930\u092e 3,8,13,18,... \u0915\u093e 15\u0935\u093e\u0901 \u092a\u0926?<\/b><br>\na3 d5 tn=3+(n-1)5 \u2192 t15=73 \u2705<br><br>\n\n<b>2. 21,18,15,... \u0915\u093e \u0915\u094c\u0928 \u0938\u093e \u092a\u0926 -81 \u0939\u0948?<\/b><br>\na21 d-3 24-3n=-81 n35 \u2705 35\u0935\u093e\u0901<br><br>\n\n<b>3. AP 11,8,5,2,... \u0915\u093e n\u0935\u093e\u0901 \u092a\u0926?<\/b><br>\ntn=14-3n \u2705<br><br>\n\n<b>4. 2-\u0905\u0902\u0915\u0940\u092f \u0915\u093f\u0924\u0928\u0940 \u0938\u0902\u0916\u094d\u092f\u093e\u090f\u0901 3 \u0938\u0947 \u0935\u093f\u092d\u093e\u091c\u094d\u092f? \u092f\u094b\u0917?<\/b><br>\n12..99 30 \u0938\u0902\u0916\u094d\u092f\u093e\u090f\u0901 \u092f\u094b\u0917 1665 \u2705<br><br>\n\n<b>5. GP 5,25,125... \u0915\u093e 6\u0935\u093e\u0901 \u092a\u0926?<\/b><br>\n5^6=15625 \u2705<br><br>\n\n<b>6. GP 2,6,18... \u0915\u093e \u0915\u094c\u0928 \u0938\u093e \u092a\u0926 4374?<\/b><br>\n2\u00d73^(n-1)=4374 n8 \u2705<br>\n\n<\/div><\/div>\n\n<div id=\"ag8-2\" style=\"display:none;\"><div style=\"background:#1565c0;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udd35 \u092d\u093e\u0917 2\/4 - Q7-Q12<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:14px;line-height:1.9;height:560px;overflow-y:auto;\">\n\n<b>7. \u092a\u094d\u0930\u0925\u092e n \u092a\u094d\u0930\u093e\u0915\u0943\u0924 \u092f\u094b\u0917 Sn?<\/b><br>\nn(n+1)\/2 \u2705 S100=5050<br><br>\n\n<b>8. 1+2+...+25?<\/b><br>\n25\u00d726\/2=325 \u2705<br><br>\n\n<b>9. \u0924\u094d\u0930\u093f\u092d\u0941\u091c\u093e\u0915\u093e\u0930 10\u0935\u0940\u0902 \u0938\u0902\u0916\u094d\u092f\u093e?<\/b><br>\n10\u00d711\/2=55 \u2705<br><br>\n\n<b>10. \u0935\u0930\u094d\u0917 1,4,9,16 \u0905\u0917\u0932\u093e?<\/b><br>\n25 \u2705 n\u00b2<br><br>\n\n<b>11. \u092b\u093f\u092c\u094b\u0928\u093e\u091a\u0940 1,2,3,5,8 \u0905\u0917\u0932\u093e \u0926\u094b?<\/b><br>\n13,21 \u2705<br><br>\n\n<b>12. AP 3,8,13... d?<\/b><br>\n5 \u2705<br>\n\n<\/div><\/div>\n\n<div id=\"ag8-3\" style=\"display:none;\"><div style=\"background:#2e7d32;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udfe2 \u092d\u093e\u0917 3\/4 - Q13-Q18<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:14px;line-height:1.9;height:560px;overflow-y:auto;\">\n\n<b>13. 3-\u0905\u0902\u0915\u0940\u092f \u0915\u093f\u0924\u0928\u0940 7 \u0938\u0947 \u0935\u093f\u092d\u093e\u091c\u094d\u092f?<\/b><br>\n105..994 128 \u2705<br><br>\n\n<b>14. 4 \u0915\u0947 \u0917\u0941\u0923\u091c 10-250 \u092c\u0940\u091a \u0915\u093f\u0924\u0928\u0947?<\/b><br>\n12..248 60 \u2705<br><br>\n\n<b>15. \u091c\u0940\u0935\u093e\u0923\u0941 30 r2 4 \u0918\u0902\u091f\u0947 \u092c\u093e\u0926?<\/b><br>\n30\u00d72^4=480 \u2705<br><br>\n\n<b>16. GP 2,8,32... r?<\/b><br>\n4 \u2705<br><br>\n\n<b>17. AP \u092e\u0947\u0902 a+10d=38 a+15d=73 d?<\/b><br>\n7 \u2705 t31=178<br><br>\n\n<b>18. 25..58 \u092f\u094b\u0917?<\/b><br>\nS58-S24=1711-300=1411 \u2705<br>\n\n<\/div><\/div>\n\n<div id=\"ag8-4\" style=\"display:none;\"><div style=\"background:#ff6f00;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udfe0 \u092d\u093e\u0917 4\/4 - Q19-Q20 \u091f\u094d\u0930\u093f\u0915<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:14px;line-height:2;height:560px;overflow-y:auto;\">\n\n<b>19. \u092a\u0941\u0928\u0930\u093e\u0935\u0930\u094d\u0924\u0940 t1=2 tn+1=3tn-2 7\u0935\u093e\u0901 \u092a\u0926 730? \u0938\u0924\u094d\u092f?<\/b><br>\n2,4,10,28,82,244,730 \u0939\u093e\u0901 \u2705<br><br>\n\n<b>20. \u0938\u093f\u0930\u092a\u093f\u0928\u094d\u0938\u094d\u0915\u0940 \u0915\u093e\u0930\u092a\u0947\u091f r?<\/b><br>\n\u0938\u0902\u0916\u094d\u092f\u093e GP r8 tn8^n, \u0915\u094d\u0937\u0947\u0924\u094d\u0930\u092b\u0932 r8\/9 sn(8\/9)^n \u2705<br><br>\n\n<div style=\"background:#fff9c4;border:3px solid #fbc02d;padding:10px;border-radius:10px;text-align:center;font-weight:bold;\">\n\u2705 \u0905\u0917\u094d\u0928\u093f\u0935\u0940\u0930 \u091f\u094d\u0930\u093f\u0915<br>\nAP tn=a+(n-1)d Sn=n\/2(a+l)<br>\nGP tn=ar^(n-1)<br>\n\u0905\u0902\u0924\u0930 \u0938\u092e\u093e\u0928 AP, \u0905\u0928\u0941\u092a\u093e\u0924 \u0938\u092e\u093e\u0928 GP<br>\n1..n=n(n+1)\/2<br>\n2-\u0905\u0902\u0915\u0940\u092f 3 \u0938\u0947 30 \u092f\u094b\u09171665<br>\n3-\u0905\u0902\u0915\u0940\u092f 7 \u0938\u0947 128<br>\n\u092b\u093f\u092c\u094b\u0928\u093e\u091a\u0940 \u092f\u094b\u0917 \u0935\u093e\u0932\u093e<br>\n\u0905\u0917\u094d\u0928\u093f\u0935\u0940\u0930 \u092e\u0947\u0902 2 \u092a\u094d\u0930\u0936\u094d\u0928 \u092a\u0915\u094d\u0915\u093e \u0907\u0938\u0940 \u0938\u0947\n<\/div>\n\n<\/div><\/div>\n<\/div>\n\n<div style=\"background:#0a5c4b;padding:12px;display:flex;justify-content:space-between;align-items:center;\">\n<button type=\"button\" id=\"ag8p\" style=\"background:#ffcc00;color:#000;border:3px solid #fff;border-radius:30px;padding:10px 22px;font-weight:bold;cursor:pointer;min-width:100px;\">&larr; \u092a\u0940\u091b\u0947<\/button>\n<span id=\"ag8i\" style=\"background:#fff;color:#0a5c4b;padding:8px 20px;border-radius:20px;font-weight:bold;\">1 \/ 4<\/span>\n<button type=\"button\" id=\"ag8n\" style=\"background:#00e676;color:#000;border:3px solid #fff;border-radius:30px;padding:10px 22px;font-weight:bold;cursor:pointer;min-width:100px;\">\u0906\u0917\u0947 &rarr;<\/button>\n<\/div>\n<\/div>\n\n<script>\n(function(){\nvar c=1,t=4;\nfunction s(n){\nfor(var i=1;i<=t;i++){\nvar el=document.getElementById('ag8-'+i);\nif(el)el.style.display=(i===n?'block':'none');\n}\ndocument.getElementById('ag8i').textContent=n+' \/ '+t;\nc=n;\n}\ndocument.getElementById('ag8n').onclick=function(){var n=c+1;if(n>t)n=1;s(n);};\ndocument.getElementById('ag8p').onclick=function(){var n=c-1;if(n<1)n=t;s(n);};\n})();\n<\/script><\/div><\/div><\/div><\/div><\/div>","protected":false},"excerpt":{"rendered":"<p>\u0905\u0927\u094d\u092f\u093e\u092f 8 &#8211; \u0905\u0928\u0941\u0915\u094d\u0930\u092e \u0914\u0930 \u0936\u094d\u0930\u0947\u0922\u093c\u0940 &#8211; \u0938\u093e\u092e\u093e\u0928\u094d\u092f \u092a\u0930\u093f\u091a\u092f \u0936\u0941\u0926\u094d\u0927 \ud83d\udd34 \u092d\u093e\u0917 1\/5 &#8211; \u0905\u0928\u0941\u0915\u094d\u0930\u092e \u0915\u094d\u092f\u093e \u0939\u0948? 8 | \u0905\u0917\u0932\u0947 \u092a\u0926 \u0915\u093e \u092a\u0942\u0930\u094d\u0935\u093e\u0928\u0941\u092e\u093e\u0928 \u2014 \u0905\u0928\u0941\u0915\u094d\u0930\u092e\u094b\u0902 \u0914\u0930 \u0936\u094d\u0930\u0947\u0922\u093c\u093f\u092f\u094b\u0902 \u0915\u093e \u0905\u0928\u094d\u0935\u0947\u0937\u0923 \u0905\u0928\u0941\u0915\u094d\u0930\u092e \u0915\u094d\u092f\u093e \u0939\u0948? \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0940 \u0910\u0938\u0940 \u0938\u0942\u091a\u0940 \u091c\u094b \u0915\u093f\u0938\u0940 \u0928\u093f\u0936\u094d\u091a\u093f\u0924 \u0928\u093f\u092f\u092e \u0915\u0947 \u0905\u0928\u0941\u0938\u093e\u0930 \u0935\u094d\u092f\u0935\u0938\u094d\u0925\u093f\u0924 \u0939\u094b, \u0905\u0928\u0941\u0915\u094d\u0930\u092e \u0915\u0939\u0932\u093e\u0924\u0940 \u0939\u0948\u0964 \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u0938\u0902\u0916\u094d\u092f\u093e \u0915\u094b \u0905\u0928\u0941\u0915\u094d\u0930\u092e \u0915\u093e \u092a\u0926 \u0915\u0939\u093e \u091c\u093e\u0924\u093e [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[10],"tags":[],"class_list":["post-991","post","type-post","status-publish","format-standard","hentry","category-mathematics"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.5 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Mathematics &#187; *8* | \u0905\u0917\u0932\u0947 \u092a\u0926 \u0915\u093e \u092a\u0942\u0930\u094d\u0935\u093e\u0928\u0941\u092e\u093e\u0928 \u2014 \u0905\u0928\u0941\u0915\u094d\u0930\u092e\u094b\u0902 \u0914\u0930 \u0936\u094d\u0930\u0947\u0922\u093c\u093f\u092f\u094b\u0902 \u0915\u093e \u0905\u0928\u094d\u0935\u0947\u0937\u0923 | Predicting the Next Term \u2014 Exploration of Sequences and Series &#187; Class-9th - Agniveer Foundation Hub | Pargai Defence &amp; Academic.<\/title>\n<meta name=\"description\" content=\"Class 9 Online Classes in Bageshwar with NCERT Textbook based study material, chapter-wise learning, MCQs, assignments and Agniveer Foundation. 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