{"id":12483,"date":"2021-05-27T15:28:26","date_gmt":"2021-05-27T09:58:26","guid":{"rendered":"https:\/\/learnsteer.sasnaka.org\/science\/?p=12483"},"modified":"2021-11-23T07:22:01","modified_gmt":"2021-11-23T01:52:01","slug":"02-09-01","status":"publish","type":"post","link":"https:\/\/learnsteer.sasnaka.org\/science\/advanced-level-science\/02-09-01\/","title":{"rendered":"02.09.01. &#8211; \u0dc1\u0dca\u200d\u0dbb\u0dda\u0dab\u0dd2 \u0db4\u0dd2\u0dbd\u0dd2\u0db6\u0db3 \u0db8\u0dd4\u0dbd\u0dd2\u0d9a \u0dc3\u0d82\u0d9a\u0dbd\u0dca\u0db4"},"content":{"rendered":"\r\n<div class=\"wp-block-buttons is-content-justification-center is-layout-flex wp-block-buttons-is-layout-flex\">\r\n<div class=\"wp-block-button is-style-shadow\"><a class=\"wp-block-button__link\" style=\"border-radius: 15px\" href=\"https:\/\/drive.google.com\/uc?id=1t9Q8g-Mwi6Zt4Lxi-_UjJiQbkR9IIA07&amp;export=download\" target=\"_blank\" rel=\"noreferrer noopener\">\u0db4\u0dcf\u0da9\u0db8\u0dda \u0dc3\u0da7\u0dc4\u0db1 Download \u0d9a\u0dbb\u0d9c\u0db1\u0dca\u0db1.<\/a><\/div>\r\n<\/div>\r\n\r\n\r\n\r\n<p><strong><span class=\"has-inline-color has-vivid-red-color\">\u0dc3\u0d82\u0dba\u0dd4\u0d9a\u0dca\u0dad \u0d9c\u0dab\u0dd2\u0dad\u0dba I (\u0dc1\u0dd4\u0daf\u0dca\u0db0 \u0d9c\u0dab\u0dd2\u0dad\u0dba) \u0db4\u0dca\u200d\u0dbb\u0dc1\u0dca\u0db1 \u0db4\u0dad\u0dca\u200d\u0dbb\u0dba\u0dda B \u0d9a\u0ddc\u0da7\u0dc3\u0dda (\u0dbb\u0da0\u0db1\u0dcf \u0db4\u0dca\u200d\u0dbb\u0dc1\u0dca\u0db1) 12 \u0dc0\u0dd0\u0db1\u0dd2 \u0d9c\u0dd0\u0da7\u0dbd\u0dd4\u0dc0\u0dd9\u0dc4\u0dd2 (b) \u0d9a\u0ddc\u0da7\u0dc3\u0dda \u0d85\u0da9\u0d82\u0d9c\u0dd4 \u0dc0\u0db1\u0dca\u0db1\u0dda \u0db8\u0dd9\u0db8 \u0db4\u0dcf\u0da9\u0db8\u0dda \u0d85\u0da9\u0d82\u0d9c\u0dd4 \u0dc3\u0dd2\u0daf\u0dca\u0db0\u0dcf\u0db1\u0dca\u0dad \u0dc0\u0dda.<\/span><\/strong><\/p>\r\n\r\n\r\n\r\n<div class=\"wp-block-media-text is-stacked-on-mobile has-background\" style=\"background-color: #bdc28d;grid-template-columns: 31% auto\">\r\n<figure class=\"wp-block-media-text__media\"><img loading=\"lazy\" decoding=\"async\" width=\"1200\" height=\"1529\" class=\"wp-image-33590 size-full\" src=\"https:\/\/learnsteer.sasnaka.org\/science\/wp-content\/uploads\/sites\/3\/2021\/10\/carl-3-1.jpg\" alt=\"\" \/><\/figure>\r\n<div class=\"wp-block-media-text__content\">\r\n<p>\u0dc1\u0dca\u200d\u0dbb\u0dda\u0dab\u0dd2 \u0db4\u0dd2\u0dc5\u0dd2\u0db6\u0daf \u0dc3\u0d82\u0d9a\u0dbd\u0dca\u0db4\u0dba \u0d89\u0dad\u0dcf \u0d88\u0dad \u0d85\u0dad\u0dd3\u0dad\u0dba\u0da7 \u0daf\u0dd2\u0dc0 \u0dba\u0db1\u0dca\u0db1\u0d9a\u0dca.<\/p>\r\n\r\n\r\n\r\n<p>Carl Friedrich Gauss (1777-1855) \u0dba\u0db1\u0dd4 \u00a0\u0da2\u0dbb\u0dca\u0db8\u0dcf\u0db1\u0dd4 \u0da2\u0dcf\u0dad\u0dd2\u0d9a\u00a0 \u0d9c\u0dab\u0dd2\u0dad\u0da5\u0dba\u0dd9\u0d9a\u0dca. \u0d9a\u0dd4\u0da9\u0dcf \u0d9c\u0dc0\u0dd4\u0dc3\u0dca \u0da7 \u0db4\u0dcf\u0dc3\u0dd0\u0dbd\u0dda\u0daf\u0dd3 \u0dad\u0dcf\u0db8 \u0d9c\u0dd4\u0dbb\u0dd4\u0dc0\u0dbb\u0dd2\u0dba \u0dc0\u0dd2\u0dc3\u0dd2\u0db1\u0dca \u0daf\u0dd9\u0db1\u0dd4 \u0dbd\u0dd0\u0db6\u0dd6 \u0db4\u0dd1\u0dc0\u0dbb\u0dd4\u0db8\u0d9a\u0da7 \u0db4\u0dd2\u0dc5\u0dd2\u0dad\u0dd4\u0dbb\u0dd4 \u0dc3\u0dd9\u0dc0\u0dd3\u0db8\u0dda\u0daf\u0dd3\u00a0 \u0db8\u0dd9\u0db8 \u0dc1\u0dca\u200d\u0dbb\u0dda\u0dab\u0dd2\u0dc0\u0dbd \u0d93\u0d9a\u0dca\u200d\u0dba\u0dba \u0dc3\u0dd9\u0dc0\u0dd3\u0db8\u0dda \u0dc3\u0dd2\u0daf\u0dca\u0db0\u0dcf\u0db1\u0dca\u0dad \u0db6\u0dd2\u0dc4\u0dd2 \u0dc0\u0dd6\u0dc0\u0dcf \u0dba\u0dd0\u0dba\u0dd2 \u0dc3\u0dd0\u0dbd\u0d9a\u0dda. \u0d9c\u0dd4\u0dbb\u0dd4\u0dad\u0dd4\u0db8\u0dd2\u0dba \u0d9c\u0dda \u0d85\u0db7\u0dd2\u0dba\u0ddd\u0d9c\u0dba \u0dc0\u0dd4\u0dab\u0dda 1 \u0dc3\u0dd2\u0da7 100 \u0daf\u0d9a\u0dca\u0dc0\u0dcf \u0dc3\u0d82\u0d9b\u0dca\u200d\u0dba\u0dcf \u0dc0\u0dbd \u0d91\u0d9a\u0dad\u0dd4\u0dc0 \u0dc3\u0dd9\u0dc0\u0dd3\u0db8\u0dba\u0dd2.<\/p>\r\n<\/div>\r\n<\/div>\r\n\r\n\r\n\r\n<div class=\"wp-block-cover alignfull has-background-dim\" style=\"background-color: #bdc28d;margin-top: 0px\">\r\n<div class=\"wp-block-cover__inner-container is-layout-flow wp-block-cover-is-layout-flow\">\r\n<p class=\"has-black-color has-text-color has-background\" style=\"background-color: #bdc28d;margin-top: -12px\">\u0d9c\u0dc0\u0dd4\u0dc3\u0dca\u0d9c\u0dda \u0db4\u0dd2\u0dc5\u0dd2\u0dad\u0dd4\u0dbb:<br \/>\u0db4\u0daf \u0dc3\u0dd2\u0dba\u0dbd\u0dca\u0dbd\u0dda\u0db8 \u0d91\u0d9a\u0dad\u0dd4\u0dc0 S \u0d9a\u0dd2\u0dba\u0dbd\u0dcf \u0dc4\u0dd2\u0dad\u0db8\u0dd4.<\/p>\r\n\r\n\r\n\r\n<p class=\"has-black-color has-text-color has-background\" style=\"background-color: #bdc28d;margin-top: -12px\">\u0db8\u0dd4\u0dbd\u0dd2\u0db1\u0dca\u0db8 1 \u0dc3\u0dd2\u0da7 50 \u0daf\u0d9a\u0dca\u0dc0\u0dcf \u0dc3\u0d82\u0d9b\u0dca\u200d\u0dba\u0dcf \u0dc0\u0dbd \u0d91\u0d9a\u0dad\u0dd4\u0dc0 \u0db4\u0dd2\u0dc5\u0dd2\u0db6\u0daf\u0dc0 \u0d85\u0dc0\u0db0\u0dcf\u0db1\u0dba \u0dba\u0ddc\u0db8\u0dd4 \u0d9a\u0dbb\u0db8\u0dd4. \u0d91\u0dba \u0db4\u0dc4\u0dad \u0d86\u0d9a\u0dcf\u0dbb\u0dba\u0da7 \u0dbd\u0dd2\u0dba\u0db1\u0dca\u0db1 \u0db4\u0dd4\u0dc5\u0dd4\u0dc0\u0db1\u0dca \u0db1\u0dda\u0daf? \u0d91\u0dba S<sub>1<\/sub> \u0d9a\u0dd2\u0dba\u0dbd \u0dc4\u0dd2\u0dad\u0db8\u0dd4.<\/p>\r\n\r\n\r\n\r\n<p class=\"has-text-align-center has-black-color has-text-color has-background\" style=\"background-color: #bdc28d;margin-top: -12px\">S<sub>1<\/sub>=1+2+3+4+\u2026+49+50 &#8212;&#8212;&#8212;&#8212;&#8211;(1)<\/p>\r\n\r\n\r\n\r\n<p class=\"has-black-color has-text-color has-background\" style=\"background-color: #bdc28d;margin-top: -12px\">\u0daf\u0dd0\u0db1\u0dca 51 \u0dc3\u0dd2\u0da7 100 \u0daf\u0d9a\u0dca\u0dc0\u0dcf \u0dc3\u0d82\u0d9b\u0dca\u200d\u0dba\u0dcf \u0dad\u0dca \u0db8\u0dda \u0dc0\u0dd2\u0daf\u0dd2\u0dba\u0da7\u0db8 \u0dc4\u0dd0\u0db6\u0dd0\u0dba\u0dd2 \u0d85\u0dc0\u0dbb\u0ddd\u0dc4\u0dab \u0db4\u0da7\u0dd2\u0db4\u0dcf\u0da7\u0dd2\u0dba\u0da7 \u0dbd\u0dd2\u0dba\u0dcf \u0d9c\u0db1\u0dca\u0db1 \u0d95\u0db1.\u0d91\u0dad\u0d9a\u0ddc\u0da7 \u0dbd\u0dd0\u0db6\u0dd9\u0db1 \u0d85\u0d9c\u0dba\u0dad\u0dca S<sub>2<\/sub> \u0d9a\u0dd2\u0dba\u0dbd\u0dcf \u0dc4\u0dd2\u0dad\u0db8\u0dd4.<\/p>\r\n\r\n\r\n\r\n<p class=\"has-text-align-center has-black-color has-text-color has-background\" style=\"background-color: #bdc28d;margin-top: -12px\">S<sub>2<\/sub> =100+99+98+\u2026.+52+51 &#8212;&#8212;&#8212;&#8212;-(2)<\/p>\r\n\r\n\r\n\r\n<p class=\"has-black-color has-text-color has-background\" style=\"background-color: #bdc28d;margin-top: -12px\">\u0daf\u0dd0\u0db1\u0dca 1 \u0dc4\u0dcf 2 \u0dc3\u0db8\u0dd3\u0d9a\u0dbb\u0dab \u0d91\u0d9a\u0dad\u0dd4 \u0d9a\u0dbb\u0db8\u0dd4.\u0db8\u0dd9\u0dad\u0db1\u0daf\u0dd2 \u0dc3\u0dd2\u0daf\u0dd4 \u0dc0\u0dd9\u0db1 \u0daf\u0dda \u0db4\u0dda\u0db1\u0dc0\u0dcf \u0db1\u0dda\u0daf? \u0dc4\u0dd0\u0db8 \u0db4\u0daf\u0dba\u0d9a\u0dca \u0db8 101 \u0dc0\u0dd9\u0db1\u0dc0\u0dcf. \u0d94\u0d9a\u0dca\u0d9a\u0ddc\u0db8 \u0db4\u0daf 50 \u0dbd\u0dd0\u0db6\u0dd9\u0db1\u0dc0\u0dcf. \u0daf\u0dd0\u0db1\u0dca \u0db8\u0dd4\u0dc5\u0dd4 \u0d91\u0d9a\u0dad\u0dd4\u0dc0 (S) 101\u00d750 \u0da7 \u0dc3\u0db8\u0dcf\u0db1 \u0db6\u0dc0 \u0db4\u0dd9\u0db1\u0dd9\u0db1\u0dc0\u0dcf \u0db1\u0dda\u0daf?<\/p>\r\n\r\n\r\n\r\n<p class=\"has-text-align-center has-black-color has-text-color has-background\" style=\"background-color: #bdc28d;margin-top: -12px\"><span class=\"wp-katex-eq\" data-display=\"false\">\u0dbd\u0dd0\u0db6\u0dd4\u0dab\\;\u0db4\u0daf\\;\u0d9c\u0dab\u0db1(50)\\;=\\;\\frac{100(\u0dc1\u0dca\u200d\u0dbb\u0dda\u0dab\u0dd2\u0dba\u0dda\\;\u0db8\u0dd4\u0dc5\u0dd4\\;\u0db4\u0daf\\;\u0d9c\u0dab\u0db1)}2<\/span><\/p>\r\n\r\n\r\n\r\n<p class=\"has-black-color has-text-color has-background\" style=\"background-color: #bdc28d;margin-top: -12px\">\u0d85\u0db4\u0dd2 \u0dc4\u0dd0\u0db8\u0ddd\u0db8 \u0daf\u0db1\u0dca\u0db1, \u0dc3\u0db8\u0dcf\u0db1\u0dca\u0dad\u0dbb \u0dc1\u0dca\u200d\u0dbb\u0dda\u0daa\u0dd2\u0dba\u0d9a \u0db4\u0daf n\u00a0 \u0d9c\u0dab\u0db1\u0d9a \u0d93\u0d9a\u0dca\u200d\u0dba\u0dba\u00a0 \u0dc3\u0ddc\u0dba\u0db1\u0dca\u0db1 \u0db7\u0dcf\u0dc0\u0dd2\u0dad\u0dcf \u00a0\u0d9a\u0dbb\u0db1\u00a0<span class=\"wp-katex-eq\" data-display=\"false\">S=\\frac{n(n+1)}2<\/span> \u0dc3\u0dd6\u0dad\u0dca\u200d\u0dbb\u0dba \u0db6\u0dd2\u0dc4\u0dd2\u0dc0\u0dd4\u0dab\u0dda \u0d92 \u0dc0\u0dd2\u0daf\u0dd2\u0dba\u0da7.<\/p>\r\n<\/div>\r\n<\/div>\r\n\r\n\r\n\r\n<ul class=\"wp-block-list\">\r\n<li>\u0db4\u0dd2\u0dc5\u0dd2\u0dc0\u0dd9\u0dbd\u0d9a\u0da7 \u0dc3\u0d9a\u0dc3\u0db1 \u0dbd\u0daf \u0dc3\u0d82\u0d9b\u0dca\u200d\u0dba\u0dcf \u0d85\u0db1\u0dd4\u0db4\u0dd2\u0dc5\u0dd2\u0dc0\u0dd9\u0dbd\u0d9a \u0d91\u0d9a\u0dad\u0dd4\u0dc0 \u0dc1\u0dca\u200d\u0dbb\u0dda\u0dab\u0dd2\u0dba\u0d9a\u0dca \u0db1\u0db8\u0dca \u0dc0\u0dda.<\/li>\r\n<li>\u0dc1\u0dca\u200d\u0dbb\u0dda\u0dab\u0dd2\u0dba\u0d9a \u0db4\u0daf \u0d9c\u0dab\u0db1 \u0db4\u0dbb\u0dd2\u0db8\u0dd2\u0dad \u0dc0\u0dda \u0db1\u0db8\u0dca \u0d91\u0dc0\u0dd0\u0db1\u0dd2 \u0dc1\u0dca\u200d\u0dbb\u0dda\u0dab\u0dd2 \u0db4\u0dbb\u0dd2\u0db8\u0dd2\u0dad \u0d91\u0d9a\u0dad\u0dd4\u0dc0\u0d9a\u0dca \u0dc3\u0dc4\u0dd2\u0dad \u0dc1\u0dca\u200d\u0dbb\u0dda\u0dab\u0dd2 \u0dbd\u0dd9\u0dc3 \u0dc3\u0dd0\u0dbd\u0d9a\u0dda.<\/li>\r\n<\/ul>\r\n\r\n\r\n\r\n<p><strong>\u0dc3\u0dd2\u0d9c\u0dca\u0db8\u0dcf \u0d85\u0d82\u0d9a\u0db1\u0dba<\/strong><\/p>\r\n\r\n\r\n\r\n<ul class=\"wp-block-list\">\r\n<li>\u0d91\u0d9a\u0dad\u0dd4\u0dc0\u0d9a\u0dca \u0dc4\u0d9a\u0dd4\u0dbd\u0dcf \u0dbd\u0dd2\u0dc0\u0dd3\u0db8\u0da7 \u0dc3\u0dd2\u0d9c\u0dca\u0db8\u0dcf \u0d85\u0d82\u0d9a\u0db1\u0dba \u0db7\u0dcf\u0dc0\u0dd2\u0dad\u0dcf \u0d9a\u0dbb\u0dba\u0dd2.<\/li>\r\n<\/ul>\r\n\r\n\r\n\r\n<span class=\"wp-katex-eq\" data-display=\"false\">\\sum_{r=1}^nU_r=U_1+U_2+U_3+\u2026+U_n<\/span>\r\n\r\n\r\n\r\n<p>\u0d8b\u0daf\u0dcf:<\/p>\r\n\r\n\r\n\r\n<span class=\"wp-katex-eq\" data-display=\"false\">\\sum_{r=1}^5r^2=1^2+2^2+3^2+4^2+5^2<\/span>\r\n\r\n\r\n\r\n<p><strong><span class=\"has-inline-color\" style=\"color: #304170\">\u0dc3\u0dd2\u0d9c\u0dca\u0db8\u0dcf \u0d85\u0d82\u0d9a\u0db1\u0dba\u0dda \u0d9c\u0dd4\u0dab<\/span><\/strong><\/p>\r\n\r\n\r\n\r\n<p>U<sub>r<\/sub> \u0dc4\u0dcf V<sub>r<\/sub> \u0dba\u0db1\u0dd4 r \u0dc4\u0dd2 \u0dc1\u0dca\u200d\u0dbb\u0dd2\u0dad \u0daf\u0dd9\u0d9a\u0d9a\u0dca \u0db1\u0db8\u0dca,<\/p>\r\n\r\n\r\n\r\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{l}1.\\sum_{r=1}^n\\left(U_r+V_r\\right)=\\sum_{r=1}^nU_r+\\sum_{r=1}^nV_r\\\\\\\\2.\\sum_{r=1}^n\\left(U_r-V_r\\right)=\\sum_{r=1}^nU_r-\\sum_{r=1}^nV_r\\\\\\\\3.\\sum_{r=1}^nkU_r=k\\sum_{r=1}^nU_r\\;\\text{(k\u00a0\u0db1\u0dd2\u0dba\u0dad\u0dba\u0d9a\u0dca\u00a0\u0dc0\u0dd2\u0da7)}\\\\\\\\4.\\sum_{r=1}^nU_rV_r\\neq\\sum_{r=1}^nU_r\\sum_{r=1}^nV_r\\\\\\\\5.\\sum_{r=1}^n\\frac{U_r}{V_r}\\neq\\frac{\\sum_{r=1}^nU_r}{\\sum_{r=1}^nV_r}\\end{array}<\/span>\r\n\r\n\r\n\r\n<h3 class=\"wp-block-heading\"><strong>\u0dc3\u0db8\u0dcf\u0db1\u0dca\u0dad\u0dbb \u0dc1\u0dca\u200d\u0dbb\u0dda\u0dab\u0dd2<\/strong><\/h3>\r\n\r\n\r\n\r\n<ul class=\"wp-block-list\">\r\n<li>\u0dc1\u0dca\u200d\u0dbb\u0dda\u0dab\u0dd2\u0dba\u0d9a \u0d91\u0d9a\u0dca \u0d91\u0d9a\u0dca \u0db4\u0daf\u0dba\u0dda \u0dc4\u0dcf \u0d8a\u0da7 \u0db4\u0dc3\u0dd4 \u0db4\u0daf\u0dba\u0dda \u0dc0\u0dd9\u0db1\u0dc3 \u0db1\u0dd2\u0dba\u0dad\u0dba\u0d9a\u0dca \u0db1\u0db8\u0dca \u0d91\u0dc0\u0dd0\u0db1\u0dd2 \u0dc1\u0dca\u200d\u0dbb\u0dda\u0dab\u0dd2\u0dba\u0d9a\u0dca \u0dc3\u0db8\u0dcf\u0db1\u0dca\u0dad\u0dbb \u0dc1\u0dca\u200d\u0dbb\u0dda\u0dab\u0dd2\u0dba\u0d9a\u0dca \u0dbd\u0dd9\u0dc3 \u0dc3\u0dd0\u0dbd\u0d9a\u0dda.<\/li>\r\n<li>\u0d9a\u0dd2\u0dc3\u0dd2\u0dba\u0db8\u0dca \u0dc3\u0db8\u0dcf\u0db1\u0dca\u0dad\u0dbb \u0dc1\u0dca\u200d\u0dbb\u0dda\u0dab\u0dd2\u0dba\u0d9a \u0db4\u0dc5\u0db8\u0dd4 \u0db4\u0daf\u0dba a \u0daf \u0db4\u0ddc\u0daf\u0dd4 \u0d85\u0db1\u0dca\u0dad\u0dbb\u0dbad \u0daf \u0dba\u0dd0\u0dba\u0dd2 \u0d9c\u0db1\u0dd2\u0db8\u0dd4. \u0dc1\u0dca\u200d\u0dbb\u0dda\u0dab\u0dd2\u0dba\u0dda r \u0dc0\u0db1 \u0db4\u0daf\u0dba\u0da7 \u0db4\u0dca\u200d\u0dbb\u0d9a\u0dcf\u0dc1\u0db1\u0dba\u0d9a\u0dca \u0db4\u0dc4\u0dad \u0db4\u0dbb\u0dd2\u0daf\u0dd2 \u0d9c\u0dad \u0dc4\u0dd0\u0d9a.<\/li>\r\n<\/ul>\r\n\r\n\r\n\r\n<p><em>\u00a0\u00a0 <\/em>\u0db4\u0dc5\u0db8\u0dd4 \u0db4\u0daf\u0dba =<em> \u00a0<\/em>a \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0= a+(1-1)d<\/p>\r\n\r\n\r\n\r\n<p><em>\u00a0<\/em>\u0daf\u0dd9\u0dc0\u0db1 \u0db4\u0daf\u0dba =<em> \u00a0<\/em>a+d \u00a0\u00a0= a+(2-1)d<\/p>\r\n\r\n\r\n\r\n<p>\u0dad\u0dd9\u0dc0\u0db1 \u0db4\u0daf\u0dba =<em> \u00a0<\/em>a+2d = a+(3-1)d<\/p>\r\n\r\n\r\n\r\n<p>\u0dc3\u0dd2\u0dc0\u0dca\u0dc0\u0db1 \u0db4\u0daf\u0dba =<em> \u00a0<\/em>a+3d = a+(4-1)d<\/p>\r\n\r\n\r\n\r\n<p>\u00a0r \u0dc0\u0db1 \u0db4\u0daf\u0dba<em>=<\/em>= a+(r-1)d\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/p>\r\n\r\n\r\n\r\n<p class=\"has-text-align-center\"><span class=\"wp-katex-eq\" data-display=\"false\">\\therefore<\/span><span class=\"has-inline-color has-vivid-red-color\">\u0dc3\u0db8\u0dcf\u0db1\u0dca\u0dad\u0dbb \u0dc1\u0dca\u200d\u0dbb\u0dda\u0dab\u0dd2\u0dba\u0d9a \u0db4\u0ddc\u0daf\u0dd4 \u0db4\u0daf\u0dba = <strong>a+(r-1)d<\/strong><\/span><\/p>\r\n\r\n\r\n\r\n<ul class=\"wp-block-list\">\r\n<li><span class=\"has-inline-color has-vivid-red-color\">\u0db8\u0dd9\u0dba \u0dc3\u0dcf\u0db0\u0db1\u0dba \u0d9a\u0dd2\u0dbb\u0dd3\u0db8\u0d9a\u0dca \u0d85\u0dc0\u0dc1\u0dca\u200d\u0dba \u0db1\u0ddc\u0dc0\u0dda.<\/span><\/li>\r\n<\/ul>\r\n\r\n\r\n\r\n<p>\u0db4\u0dc5\u0db8\u0dd4 \u0db4\u0daf\u0dba a \u0dc4\u0dcf \u0d85\u0dc0\u0dc3\u0db1\u0dca \u0db4\u0daf\u0dba l \u0dc0\u0db1 \u0db4\u0daf n \u0db4\u0dca\u200d\u0dbb\u0db8\u0dcf\u0dab\u0dba\u0d9a\u0dca \u0d87\u0dad\u0dd2 \u0db4\u0ddc\u0daf\u0dd4 \u0d85\u0db1\u0dca\u0dad\u0dbb\u0dba d \u0dc0\u0dd6 \u0dc3\u0db8\u0dcf\u0db1\u0dca\u0dad\u0dbb \u0dc1\u0dca\u200d\u0dbb\u0dda\u0dab\u0dd2\u0dba\u0d9a \u0db4\u0daf\u0dc0\u0dbd \u0d93\u0d9a\u0dca\u200d\u0dba\u0dba S<sub>n <\/sub>\u0db1\u0db8\u0dca,<\/p>\r\n\r\n\r\n\r\n<p>S<sub>n<\/sub> = a + (a+d) + (a+2d) + \u2026. + (l-2d) + (l-d) + l <span class=\"wp-katex-eq\" data-display=\"false\">\\rightarrow<\/span>(1)<\/p>\r\n\r\n\r\n\r\n<p>S<sub>n<\/sub> = l + (l-d) + (l-2d) + \u2026. + (a+2d) + (a+d) + a <span class=\"wp-katex-eq\" data-display=\"false\">\\rightarrow<\/span>(2)<\/p>\r\n\r\n\r\n\r\n<p>(1)+(2) =&gt;<\/p>\r\n\r\n\r\n\r\n<p>2S<sub>n<\/sub> = (a+l) + (a+l) + (a+l) + \u2026. + (a+l) + (a+l) + (a+l)<\/p>\r\n\r\n\r\n\r\n<p>2S<sub>n <\/sub>= n(a+l)<\/p>\r\n\r\n\r\n\r\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}S_n&amp;=&amp;\\frac n2\\left(a+l\\right)\\\\l&amp;=&amp;a+\\left(n-1\\right)d\\;\\text{\u0db1\u0dd2\u0dc3\u0dcf,}\\\\S_n&amp;=&amp;\\frac n2\\left(a+a+\\left(n-1\\right)d\\right)\\\\S_n&amp;=&amp;\\frac n2\\left(2a+\\left(n-1\\right)d\\right)\\end{array}<\/span>\r\n\r\n\r\n\r\n<h3 class=\"wp-block-heading\"><strong>\u0d9c\u0dd4\u0dab\u0ddd\u0dad\u0dca\u0dad\u0dbb \u0dc1\u0dca\u200d\u0dbb\u0dda\u0dab\u0dd2<\/strong><\/h3>\r\n\r\n\r\n\r\n<ul class=\"wp-block-list\">\r\n<li>\u0dc1\u0dca\u200d\u0dbb\u0dda\u0dab\u0dd2\u0dba\u0d9a \u0d91\u0d9a\u0dca \u0d91\u0d9a\u0dca \u0db4\u0daf\u0dba \u0dc4\u0dcf \u0d8a\u0da7 \u0db4\u0dc3\u0dd4 \u0db4\u0daf\u0dba \u0d85\u0dad\u0dbb \u0d85\u0db1\u0dd4\u0db4\u0dcf\u0dad\u0dba \u0db1\u0dd2\u0dba\u0dad\u0dba\u0d9a\u0dca \u0db1\u0db8\u0dca \u0d91\u0dc0\u0dd0\u0db1\u0dd2 \u0dc1\u0dca\u200d\u0dbb\u0dda\u0dab\u0dd2\u0dba\u0d9a\u0dca \u0d9c\u0dd4\u0dab\u0ddd\u0dad\u0dca\u0dad\u0dbb \u0dc1\u0dca\u200d\u0dbb\u0dda\u0dab\u0dd2\u0dba\u0d9a\u0dca \u0dbd\u0dd9\u0dc3 \u0dc3\u0dd0\u0dbd\u0d9a\u0dda.<\/li>\r\n<li>\u0d9a\u0dd2\u0dc3\u0dd2\u0dba\u0db8\u0dca \u0d9c\u0dd4\u0dab\u0ddd\u0dad\u0dca\u0dad\u0dbb \u0dc1\u0dca\u200d\u0dbb\u0dda\u0dab\u0dd2\u0dba\u0d9a \u0db8\u0dd4\u0dbd\u0dca \u0db4\u0daf\u0dba a \u0dc3\u0dc4 \u0db4\u0ddc\u0daf\u0dd4 \u0d85\u0db1\u0dd4\u0db4\u0dcf\u0dad\u0dba x \u0dc0\u0dda \u0dba\u0dd0\u0dba\u0dd2 \u0d9c\u0db1\u0dd2\u0db8\u0dd4. \u0d91\u0db8 \u0dc1\u0dca\u200d\u0dbb\u0dda\u0dab\u0dd2\u0dba\u0dda r \u0dc0\u0db1 \u0db4\u0daf\u0dba \u0dc3\u0daf\u0dc4\u0dcf \u0db4\u0dca\u200d\u0dbb\u0d9a\u0dcf\u0dc1\u0db1\u0dba\u0d9a\u0dca \u0db4\u0dc4\u0dad \u0db4\u0dbb\u0dd2\u0daf\u0dd2 \u0dbd\u0db6\u0dcf \u0d9c\u0dad \u0dc4\u0dd0\u0d9a.<\/li>\r\n<\/ul>\r\n\r\n\r\n\r\n<p><em>\u00a0\u00a0 <\/em>\u0db4\u0dc5\u0db8\u0dd4 \u0db4\u0daf\u0dba =<em>\u00a0 <\/em>a\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 = ax<sup>1-1<\/sup><\/p>\r\n\r\n\r\n\r\n<p><em>\u00a0<\/em>\u0daf\u0dd9\u0dc0\u0db1 \u0db4\u0daf\u0dba =<em>\u00a0 <\/em>ax\u00a0\u00a0\u00a0\u00a0 = ax<sup>2-1<\/sup><\/p>\r\n\r\n\r\n\r\n<p>\u0dad\u0dd9\u0dc0\u0db1 \u0db4\u0daf\u0dba =<em>\u00a0 <\/em>ax<sup>2<\/sup>\u00a0\u00a0 = ax<sup>3-1<\/sup><\/p>\r\n\r\n\r\n\r\n<p>\u00a0<span class=\"wp-katex-eq\" data-display=\"false\">\\therefore<\/span>r \u0dc0\u0db1 \u0db4\u0daf\u0dba = <em>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/em>= ax<sup>r-1<\/sup><em>\u00a0\u00a0\u00a0\u00a0<\/em><\/p>\r\n\r\n\r\n\r\n<p class=\"has-text-align-center\"><span class=\"wp-katex-eq\" data-display=\"false\">\\therefore<\/span><span class=\"has-inline-color has-vivid-red-color\">\u0d9c\u0dd4\u0dab\u0ddd\u0dad\u0dca\u0dad\u0dbb \u0dc1\u0dca\u200d\u0dbb\u0dda\u0dab\u0dd2\u0dba\u0d9a \u0db4\u0ddc\u0daf\u0dd4 \u0db4\u0daf\u0dba = <strong>ax<sup>r-1<\/sup><\/strong><\/span><\/p>\r\n\r\n\r\n\r\n<ul class=\"wp-block-list\">\r\n<li>\u0db4\u0dc5\u0db8\u0dd4 \u0db4\u0daf\u0dba a \u0dc0\u0dd6 \u0db4\u0daf n \u0db4\u0dca\u200d\u0dbb\u0db8\u0dcf\u0dab\u0dba\u0d9a\u0dca \u0d87\u0dad\u0dd2 \u0db4\u0ddc\u0daf\u0dd4 \u0d85\u0db1\u0dd4\u0db4\u0dcf\u0dad\u0dba r \u0dc0\u0db1 \u0d9c\u0dd4\u0dab\u0ddd\u0dad\u0dca\u0dad\u0dbb \u0dc1\u0dca\u200d\u0dbb\u0dda\u0dab\u0dd2\u0dba\u0d9a \u0db4\u0daf\u0dc0\u0dbd \u0d93\u0d9a\u0dca\u200d\u0dba\u0dba S<sub>n <\/sub>\u0db1\u0db8\u0dca,<\/li>\r\n<\/ul>\r\n\r\n\r\n\r\n<p>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 S<sub>n<\/sub> = a+ ar + ar<sup>2<\/sup> + \u2026 + ar<sup>n-2<\/sup> + ar<sup>n-1<\/sup> <span class=\"wp-katex-eq\" data-display=\"false\">\\rightarrow<\/span>(1)<\/p>\r\n\r\n\r\n\r\n<p>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 rS<sub>n <\/sub>= ar + ar<sup>2<\/sup> + ar<sup>3 <\/sup>+ \u2026 + ar<sup>n-1<\/sup> + ar<sup>n<\/sup> <span class=\"wp-katex-eq\" data-display=\"false\">\\rightarrow<\/span>(2)<\/p>\r\n\r\n\r\n\r\n<p>(2)-(1) =&gt;<\/p>\r\n\r\n\r\n\r\n<p>\u00a0\u00a0(r-1) S<sub>n <\/sub>= ar<sup>n<\/sup> \u2013 a<\/p>\r\n\r\n\r\n\r\n<p>\u00a0 (r-1) S<sub>n <\/sub>= a(r<sup>n<\/sup> &#8211; 1)<\/p>\r\n\r\n\r\n\r\n<p>r-1<span class=\"wp-katex-eq\" data-display=\"false\">\\neq<\/span>0 \u0dc0\u0dd2\u0da7,<\/p>\r\n\r\n\r\n\r\n<span class=\"wp-katex-eq\" data-display=\"false\">S_n=\\frac{a\\left(r^n-1\\right)}{r-1}<\/span>\r\n\r\n\r\n\r\n<p>r-1 = 0 \u0dc0\u0dd2\u0da7,\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/p>\r\n\r\n\r\n\r\n<p>S<sub>n<\/sub> = a + a + a + \u2026 + a<\/p>\r\n\r\n\r\n\r\n<p><span class=\"has-inline-color has-vivid-red-color\">S<sub>n<\/sub> = na<\/span><\/p>\r\n\r\n\r\n\r\n<div class=\"wp-block-cover alignfull has-background-dim\" style=\"background-color: #f5e8c7\">\r\n<div class=\"wp-block-cover__inner-container is-layout-flow wp-block-cover-is-layout-flow\">\r\n<p class=\"has-text-align-center has-text-color\" style=\"color: #6f4c5b;font-size: 20px\"><span class=\"wp-katex-eq\" data-display=\"false\">\\frac1{1+\\sqrt2}+\\frac1{\\sqrt2+\\sqrt3}+\\frac1{\\sqrt3{+}{\\sqrt4}}+\\frac1{\\sqrt4{+}{\\sqrt5}}+\u2026+\\frac1{\\sqrt{2020}+\\sqrt{2021}}=4041<\/span><\/p>\r\n\r\n\r\n\r\n<p class=\"has-text-align-center\" style=\"font-size: 20px\"><strong><span class=\"tadv-color\" style=\"color: #6f4c5b\">\u0dc1\u0dca\u200d\u0dbb\u0dda\u0dab\u0dd2\u0dba\u0dda\u00a0\u0d93\u0d9a\u0dca\u200d\u0dba\u00a0\u0d91\u0d9a\u0dc0\u0dbb\u00a0\u0dbd\u0dd2\u0dba\u0db1\u0dca\u0db1\u00a0\u0d94\u0dba\u0dcf\u0da7\u0dad\u0dca\u00a0\u0db4\u0dd4\u0dc5\u0dd4\u0dc0\u0db1\u0dca\u0daf?<\/span><\/strong><\/p>\r\n\r\n\r\n\r\n<div class=\"wp-block-buttons is-content-justification-center is-layout-flex wp-block-buttons-is-layout-flex\">\r\n<div class=\"wp-block-button is-style-shadow\"><a class=\"wp-block-button__link has-text-color has-background\" style=\"border-radius: 10px;background: linear-gradient(135deg,#9e7777 0%,#9e7777 100%);color: #f3e4be\" href=\"https:\/\/learnsteer.sasnaka.org\/science\/advanced-level-science\/02-09-01-q1\/\" target=\"_blank\" rel=\"noreferrer noopener\">\u0db4\u0dd2\u0dc5\u0dd2\u0dad\u0dd4\u0dbb \u0db4\u0dd9\u0db1\u0dca\u0dc0\u0db1\u0dca\u0db1<\/a><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n\r\n\r\n\r\n<p>\u0d8b\u0daf\u0dcf: <span class=\"wp-katex-eq\" data-display=\"false\">\\frac4{1.2.3}\\left(\\frac13\\right)+\\frac5{2.3.4}\\left(\\frac13\\right)^2+\\frac6{3.4.5}\\left(\\frac13\\right)^3+\u2026<\/span>\u0dc1\u0dca\u200d\u0dbb\u0dda\u0dab\u0dd2\u0dba\u0dda \u0db4\u0daf n \u0dc4\u0dd2 \u0d93\u0d9a\u0dca\u200d\u0dba\u0dba \u0dc3\u0dd2\u0d9c\u0dca\u0db8\u0dcf \u0d85\u0d82\u0d9a\u0db1\u0dba\u0dd9\u0db1\u0dca \u0dbd\u0dd2\u0dba\u0dcf \u0daf\u0d9a\u0dca\u0dc0\u0db1\u0dca\u0db1.<\/p>\r\n\r\n\r\n\r\n<p>4, 5, 6, \u2026 T<sub>r<\/sub> = r+3<\/p>\r\n\r\n\r\n\r\n<p>1, 2, 3, \u2026 T<sub>r<\/sub> = r<\/p>\r\n\r\n\r\n\r\n<p>2, 3, 4, \u2026 T<sub>r<\/sub> = r+1<\/p>\r\n\r\n\r\n\r\n<p>3, 4, 5, \u2026 T<sub>r<\/sub> = r+2<\/p>\r\n\r\n\r\n\r\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{l}\\therefore U_r(\\text{r\u00a0\u0dc0\u0db1\u00a0\u0db4\u0daf\u0dba})=\\frac{r+3}{r\\left(r+1\\right)\\left(r+2\\right)}\\left(\\frac13\\right)^r\\\\\\\\\\therefore\\text{\u0db4\u0daf\u00a0n\u00a0\u0dc4\u0dd2\u00a0\u0d93\u0d9a\u0dca\u200d\u0dba\u0dba=}\\sum_{r=1}^n\\frac{r+3}{r\\left(r+1\\right)\\left(r+2\\right)}\\left(\\frac13\\right)^r\\end{array}<\/span>\r\n\r\n\r\n\r\n<h4 class=\"wp-block-heading\"><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}&amp;&amp;\\sum_{\\mathrm r=1}^{\\mathrm n}\\mathrm r\\end{array}<\/span><strong>\u0dc1\u0dca\u200d\u0dbb\u0dda\u0dab\u0dd2\u0dba\u0dda \u0d93\u0d9a\u0dca\u200d\u0dba\u0dba \u0dc3\u0dd9\u0dc0\u0dd3\u0db8<\/strong><\/h4>\r\n\r\n\r\n\r\n<p>\u0d9a\u0dca\u200d\u0dbb\u0db8\u0dba 1<\/p>\r\n\r\n\r\n\r\n<p><span class=\"wp-katex-eq\" data-display=\"false\">\\sum_{\\mathrm r=1}^{\\mathrm n}\\mathrm r=1+2+3+\u2026+\\mathrm n<\/span>(\u0db4\u0ddc\u0daf\u0dd4 \u0d85\u0db1\u0dca\u0dad\u0dbb\u0dba 1 \u0dc0\u0dd6 \u0dc3\u0db8\u0dcf\u0db1\u0dca\u0dad\u0dbb \u0dc1\u0dca\u200d\u0dbb\u0dda\u0dab\u0dd2\u0dba\u0d9a\u0dca)<\/p>\r\n\r\n\r\n\r\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}&amp;=&amp;\\frac n2\\lbrack2\\times1+(n-1)\\times1\\rbrack\\\\&amp;=&amp;\\frac n2(n+1)\\end{array}<\/span>\r\n\r\n\r\n\r\n<p>\u0d9a\u0dca\u200d\u0dbb\u0db8\u0dba 2<\/p>\r\n\r\n\r\n\r\n<p>\u0db8\u0dd9\u0db8 \u0dc3\u0dcf\u0db0\u0db1\u0dba \u0dc3\u0daf\u0dc4\u0dcf \u0db8\u0dd9\u0db8 \u0db4\u0dcf\u0da9\u0db8\u0dda\u0db8 \u0d89\u0daf\u0dd2\u0dbb\u0dd2 \u0d9a\u0ddc\u0da7\u0dc3\u0d9a\u0daf\u0dd3 \u0dba\u0dd9\u0daf\u0dd9\u0db1 \u0dc3\u0dd2\u0daf\u0dca\u0db0\u0dcf\u0db1\u0dca\u0dad\u0dba\u0d9a\u0dca \u0db7\u0dcf\u0dc0\u0dd2\u0dad\u0dcf \u0dc0\u0dda.<\/p>\r\n\r\n\r\n\r\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\mathrm r^2-\\left(\\mathrm r-1\\right)^2&amp;=&amp;2\\mathrm r-1\\\\\\sum_{\\mathrm r=1}^{\\mathrm n}\\left[\\mathrm r^2-\\left(\\mathrm r-1\\right)^2\\right]&amp;=&amp;\\sum_{\\mathrm r=1}^{\\mathrm n}\\left[2\\mathrm r-1\\right]\\end{array}<\/span>\r\n\r\n\r\n\r\n<p>f(r) = r<sup>2<\/sup> \u0dbd\u0dd9\u0dc3 \u0d9c\u0dad\u0dca \u0dc0\u0dd2\u0da7 f(r-1) = (r-1)<sup>2<\/sup><\/p>\r\n\r\n\r\n\r\n<figure class=\"wp-block-image size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-12643\" src=\"https:\/\/learnsteer.sasnaka.org\/science\/wp-content\/uploads\/sites\/3\/2021\/03\/image-86.png\" alt=\"\" width=\"322\" height=\"454\" \/><\/figure>\r\n\r\n\r\n\r\n<h4 class=\"wp-block-heading\"><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}&amp;&amp;\\sum_{\\mathrm r=1}^{\\mathrm n}\\mathrm r^2\\end{array}<\/span><strong>\u0dc1\u0dca\u200d\u0dbb\u0dda\u0dab\u0dd2\u0dba\u0dda \u0d93\u0d9a\u0dca\u200d\u0dba\u0dba \u0dc3\u0dd9\u0dc0\u0dd3\u0db8.<\/strong><\/h4>\r\n\r\n\r\n\r\n<p>\u0db8\u0dd9\u0db8 \u0dc3\u0dcf\u0db0\u0db1\u0dba \u0dc3\u0daf\u0dc4\u0dcf \u0db8\u0dd9\u0db8 \u0db4\u0dcf\u0da9\u0db8\u0dda\u0db8 \u0d89\u0daf\u0dd2\u0dbb\u0dd2 \u0d9a\u0ddc\u0da7\u0dc3\u0d9a\u0daf\u0dd3 \u0dba\u0dd9\u0daf\u0dd9\u0db1 \u0dc3\u0dd2\u0daf\u0dca\u0db0\u0dcf\u0db1\u0dca\u0dad\u0dba\u0d9a\u0dca \u0db7\u0dcf\u0dc0\u0dd2\u0dad\u0dcf \u0dc0\u0dda.<\/p>\r\n\r\n\r\n\r\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\mathrm r^3-\\left(\\mathrm r-1\\right)^3&amp;=&amp;3\\mathrm r^2-3\\mathrm r+1\\\\\\sum_{\\mathrm r=1}^{\\mathrm n}\\left[\\mathrm r^3-\\left(\\mathrm r-1\\right)^3\\right]&amp;=&amp;\\sum_{\\mathrm r=1}^{\\mathrm n}\\left(3\\mathrm r^2-3\\mathrm r+1\\right)\\end{array}<\/span>\r\n\r\n\r\n\r\n<p>f(r) = r<sup>3<\/sup> \u0dbd\u0dd9\u0dc3 \u0d9c\u0dad\u0dca \u0dc0\u0dd2\u0da7 f(r-1) = (r-1)<sup>3<\/sup><\/p>\r\n\r\n\r\n\r\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\sum_{\\mathrm r=1}^{\\mathrm n}\\left[\\mathrm f\\left(\\mathrm r\\right)-\\mathrm f\\left(\\mathrm r-1\\right)\\right]&amp;=&amp;3\\sum_{\\mathrm r=1}^{\\mathrm n}\\mathrm r^2-3\\sum_{\\mathrm r=1}^{\\mathrm n}\\mathrm r+\\sum_{\\mathrm r=1}^{\\mathrm n}1\\end{array}<\/span>\r\n\r\n\r\n\r\n<figure class=\"wp-block-image size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-12644\" src=\"https:\/\/learnsteer.sasnaka.org\/science\/wp-content\/uploads\/sites\/3\/2021\/03\/image-87.png\" alt=\"\" width=\"374\" height=\"382\" \/><\/figure>\r\n\r\n\r\n\r\n<h4 class=\"wp-block-heading\"><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}&amp;&amp;\\sum_{\\mathrm r=1}^{\\mathrm n}\\mathrm r^3\\end{array}<\/span><strong>\u0dc1\u0dca\u200d\u0dbb\u0dda\u0dab\u0dd2\u0dba\u0dda \u0d93\u0d9a\u0dca\u200d\u0dba\u0dba \u0dc3\u0dd9\u0dc0\u0dd3\u0db8.<\/strong><\/h4>\r\n\r\n\r\n\r\n<p>\u0db8\u0dd9\u0db8 \u0dc3\u0dcf\u0db0\u0db1\u0dba \u0dc3\u0daf\u0dc4\u0dcf \u0db8\u0dd9\u0db8 \u0db4\u0dcf\u0da9\u0db8\u0dda\u0db8 \u0d89\u0daf\u0dd2\u0dbb\u0dd2 \u0d9a\u0ddc\u0da7\u0dc3\u0d9a\u0daf\u0dd3 \u0dba\u0dd9\u0daf\u0dd9\u0db1 \u0dc3\u0dd2\u0daf\u0dca\u0db0\u0dcf\u0db1\u0dca\u0dad\u0dba\u0d9a\u0dca \u0db7\u0dcf\u0dc0\u0dd2\u0dad\u0dcf \u0dc0\u0dda.<\/p>\r\n\r\n\r\n\r\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\mathrm r^2\\left(\\mathrm r+1\\right)^2-\\left(\\mathrm r-1\\right)^2\\mathrm r^2&amp;=&amp;4\\mathrm r^3\\\\\\sum_{\\mathrm r=1}^{\\mathrm n}\\left[\\mathrm r^2\\left(\\mathrm r+1\\right)^2-\\left(\\mathrm r-1\\right)^2\\mathrm r^2\\right]&amp;=&amp;\\sum_{\\mathrm r=1}^{\\mathrm n}4\\mathrm r^3\\end{array}<\/span>\r\n\r\n\r\n\r\n<p>f(r) = r<sup>2<\/sup>(r+1)<sup>2<\/sup> \u0dbd\u0dd9\u0dc3 \u0d9c\u0dad\u0dca \u0dc0\u0dd2\u0da7 f(r-1) = (r-1)<sup>2<\/sup>r<sup>2<\/sup><\/p>\r\n\r\n\r\n\r\n<figure class=\"wp-block-image size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-12646\" src=\"https:\/\/learnsteer.sasnaka.org\/science\/wp-content\/uploads\/sites\/3\/2021\/03\/image-88.png\" alt=\"\" width=\"329\" height=\"436\" \/><\/figure>\r\n\r\n\r\n\r\n<p>\u0d8b\u0daf\u0dcf:<\/p>\r\n\r\n\r\n\r\n<p>1.<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}&amp;&amp;\\sum_{\\mathrm r=1}^{\\mathrm n}\\mathrm r\\left(2\\mathrm r+3\\right)\\end{array}<\/span> \u00a0\u0dc4\u0dd2\u00a0\u0d85\u0d9c\u0dba\u00a0\u0d89\u0dc4\u0dad\u00a0\u0d9c\u0ddc\u0da9\u0db1\u0dd0\u0d9c\u0dd6\u00a0\u0dc3\u0dd6\u0dad\u0dca\u200d\u0dbb\u00a0\u0db7\u0dcf\u0dc0\u0dd2\u0dad\u0dba\u0dd9\u0db1\u0dca\u00a0\u0dc3\u0ddc\u0dba\u0db1\u0dca\u0db1.<\/p>\r\n\r\n\r\n\r\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\sum_{\\mathrm r=1}^{\\mathrm n}\\mathrm r\\left(2\\mathrm r+3\\right)&amp;=&amp;\\sum_{\\mathrm r=1}^{\\mathrm n}\\left(2\\mathrm r^2+3\\mathrm r\\right)\\\\&amp;&amp;\\\\&amp;=&amp;\\sum_{\\mathrm r=1}^{\\mathrm n}2\\mathrm r^2+\\sum_{\\mathrm r=1}^{\\mathrm n}3\\mathrm r\\\\&amp;&amp;\\\\&amp;=&amp;2\\sum_{\\mathrm r=1}^{\\mathrm n}\\mathrm r^2+3\\sum_{\\mathrm r=1}^{\\mathrm n}\\mathrm r\\\\&amp;&amp;\\\\&amp;=&amp;2\\times\\frac{\\mathrm n\\left(\\mathrm n+1\\right)\\left(2\\mathrm n+1\\right)}6+3\\times\\frac{\\mathrm n\\left(\\mathrm n+1\\right)}2\\\\&amp;&amp;\\\\&amp;=&amp;\\frac{\\mathrm n\\left(\\mathrm n+1\\right)\\left(4\\mathrm n+11\\right)}6\\end{array}<\/span>\r\n\r\n\r\n\r\n<p>2.<span class=\"wp-katex-eq\" data-display=\"false\">\\sum_{\\mathrm r=1}^{\\mathrm n}2\\mathrm r\\left(\\mathrm r+1\\right)\\left(\\mathrm r+2\\right)<\/span>\u00a0\u0dc4\u0dd2\u00a0\u0d85\u0d9c\u0dba\u00a0\u0d89\u0dc4\u0dad\u00a0\u0d9c\u0ddc\u0da9\u0db1\u0dd0\u0d9c\u0dd6\u00a0\u0dc3\u0dd6\u0dad\u0dca\u200d\u0dbb\u00a0\u0db7\u0dcf\u0dc0\u0dd2\u0dad\u0dba\u0dd9\u0db1\u0dca\u00a0\u0dc3\u0ddc\u0dba\u0db1\u0dca\u0db1.<\/p>\r\n\r\n\r\n\r\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\sum_{\\mathrm r=1}^{\\mathrm n}2\\mathrm r\\left(\\mathrm r+1\\right)\\left(\\mathrm r+2\\right)\\\\&amp;&amp;\\\\&amp;=&amp;\\overset{\\mathrm n}{\\underset{\\mathrm r=1}{\\sum\\lbrack}}2\\mathrm r^3+6\\mathrm r^2+4\\mathrm r\\rbrack\\\\&amp;&amp;\\\\&amp;=&amp;\\sum_{\\mathrm r=1}^{\\mathrm n}2\\mathrm r^3+\\sum_{\\mathrm r=1}^{\\mathrm n}6\\mathrm r^2+\\sum_{\\mathrm r=1}^{\\mathrm n}4\\mathrm r\\\\&amp;&amp;\\\\&amp;=&amp;2\\sum_{\\mathrm r=1}^{\\mathrm n}\\mathrm r^3+6\\sum_{\\mathrm r=1}^{\\mathrm n}\\mathrm r^2+4\\sum_{\\mathrm r=1}^{\\mathrm n}\\mathrm r\\\\&amp;&amp;\\\\&amp;=&amp;2\\times\\frac{\\mathrm n^2\\left(\\mathrm n+1\\right)^2}4+6\\times\\frac{\\mathrm n\\left(\\mathrm n+1\\right)\\left(2\\mathrm n+1\\right)}6+4\\times\\frac{\\mathrm n\\left(\\mathrm n+1\\right)}2\\\\&amp;&amp;\\\\&amp;=&amp;\\frac{\\mathrm n\\left(\\mathrm n+1\\right)\\left(\\mathrm n+2\\right)\\left(\\mathrm n+3\\right)}2\\end{array}<\/span>\r\n\r\n\r\n\r\n<p class=\"has-text-align-center has-white-color has-text-color has-background\" style=\"background-color: #272062\"><strong><span style=\"font-family: 'book antiqua', palatino, serif\"><em><span style=\"font-size: 18pt;color: #ffffff\">&#8220;Mathematics is the art of giving the same name to different things&#8221;<\/span><\/em><\/span><\/strong><br \/><span style=\"font-size: 10pt;color: #bdbaa2;font-family: tahoma, arial, helvetica, sans-serif\">-Henri Pioncare-<\/span><\/p>\r\n\r\n\r\n\r\n<p>&nbsp;<\/p>\r\n\r\n\r\n\r\n<div class=\"wp-block-buttons is-content-justification-center is-layout-flex wp-block-buttons-is-layout-flex\">\r\n<div class=\"wp-block-button is-style-shadow td_btn_normal\"><a class=\"wp-block-button__link\" style=\"border-radius: 15px\" href=\"https:\/\/drive.google.com\/uc?id=1t9Q8g-Mwi6Zt4Lxi-_UjJiQbkR9IIA07&amp;export=download\" target=\"_blank\" rel=\"noreferrer noopener\">\u0db4\u0dcf\u0da9\u0db8\u0dda \u0dc3\u0da7\u0dc4\u0db1 Download \u0d9a\u0dbb\u0d9c\u0db1\u0dca\u0db1.<\/a><\/div>\r\n<\/div>\r\n\r\n\r\n\r\n\r\n\r\n<p><a class=\"wp-block-button__link\" style=\"border-radius: 15px\" href=\"https:\/\/drive.google.com\/drive\/folders\/1k6Rq4b413h8HRUsRKKKg_7l_6dATmlPg?usp=sharing\" target=\"_blank\" rel=\"noreferrer noopener\">\u0dad\u0dc0\u0dad\u0dca \u0db4\u0dca\u200d\u0dbb\u0dc1\u0dca\u0db1 \u0db4\u0dd9\u0db1\u0dca\u0dc0\u0db1\u0dca\u0db1.<\/a><\/p>\r\n\r\n\r\n\r\n<div class=\"wp-block-spacer\" style=\"height: 100px\" aria-hidden=\"true\">\u00a0<\/div>\r\n\r\n\r\n\r\n<p>&nbsp;<\/p>\r\n","protected":false},"excerpt":{"rendered":"<p>\u0db4\u0dd2\u0dc5\u0dd2\u0dc0\u0dd9\u0dbd\u0d9a\u0da7 \u0dc3\u0d9a\u0dc3\u0db1 \u0dbd\u0daf \u0dc3\u0d82\u0d9b\u0dca\u200d\u0dba\u0dcf \u0d85\u0db1\u0dd4\u0d9a\u0dca\u200d\u0dbb\u0db8\u0dba\u0d9a \u0d91\u0d9a\u0dad\u0dd4\u0dc0\u0d9a\u0dca \u0dc1\u0dca\u200d\u0dbb\u0dda\u0dab\u0dd2\u0dba\u0d9a\u0dca \u0dc0\u0dda.\u0db8\u0dd9\u0db8 \u0db4\u0dcf\u0da9\u0db8\u0dda\u0daf\u0dd2, \u0dc3\u0db8\u0dcf\u0db1\u0dca\u0dad\u0dbb \u0dc4\u0dcf \u0d9c\u0dd4\u0dab\u0ddd\u0dad\u0dca\u0dad\u0dbb \u0dc1\u0dca\u200d\u0dbb\u0dda\u0dab\u0dd2\u0dba\u0d9a \u0dc3\u0dcf\u0db0\u0dcf\u0dbb\u0dab \u0db4\u0daf\u0dba \u0dc4\u0db3\u0dd4\u0db1\u0dcf \u0d9c\u0dd0\u0db1\u0dd3\u0db8 , \u0db4\u0dbb\u0dd2\u0db8\u0dd2\u0dad \u0dc4\u0dcf \u0d85\u0db4\u0dbb\u0dd2\u0db8\u0dd2\u0dad \u0dc1\u0dca\u200d\u0dbb\u0dda\u0dab\u0dd2\u0dc0\u0dbd \u0db4\u0daf \u0dc0\u0dbd  \u0d93\u0d9a\u0dca\u200d\u0dba\u0dba \u0dc3\u0db3\u0dc4\u0dcf \u0dc3\u0dd6\u0dad\u0dca\u200d\u0dbb \u0dbd\u0dd2\u0dc0\u0dd3\u0db8 \u0d85\u0db0\u0dca\u200d\u0dba\u0dba\u0db1\u0dba \u0d9a\u0dbb\u0dba\u0dd2.<\/p>\n","protected":false},"author":62,"featured_media":16553,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"tdm_status":"","tdm_grid_status":"","footnotes":""},"categories":[3659,3633,42,3630,3629],"tags":[4209,4210,4212,4211,4204,4203,4206,4207,4197,4200,4198,4196,4201,4199,4189,4190,4188,4191,4214,4187,4182,4183,4184,4185,4208,4202,4205,4186,4192,4213,4193,4194,4195,4181],"class_list":{"0":"post-12483","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-02-09-series","8":"category-02-algebra","9":"category-advanced-level-science","10":"category-pure-mathematics","11":"category-combined-mathematics","12":"tag-gunoththara-sheni","13":"tag-gunoththara-sheniya","14":"tag-gunottr-sheni","15":"tag-gunottr-sheniya","16":"tag-podu-antaraya","17":"tag-podu-antharaya","18":"tag-podu-padaya","19":"tag-podu-padhaya","20":"tag-samaantara-sheniya","21":"tag-samaanthara-sheniua","22":"tag-samantara-sheeniya","23":"tag-samantara-sheniya","24":"tag-samanthara-sheeniya","25":"tag-samanthara-sheniya","26":"tag-sheeni","27":"tag-sheeniya","28":"tag-sheni","29":"tag-sheniya","30":"tag-sheniye-aikyaya","31":"tag-shreni","32":"tag-sigma","33":"tag-sigma-ankanaya","34":"tag-zigma","35":"tag-zigma-ankanaya","36":"tag-4208","37":"tag-4202","38":"tag-4205","39":"tag-4186","40":"tag-4192","41":"tag-4213","42":"tag-4193","43":"tag-4194","44":"tag-4195","45":"tag-4181"},"_links":{"self":[{"href":"https:\/\/learnsteer.sasnaka.org\/science\/wp-json\/wp\/v2\/posts\/12483"}],"collection":[{"href":"https:\/\/learnsteer.sasnaka.org\/science\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/learnsteer.sasnaka.org\/science\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/learnsteer.sasnaka.org\/science\/wp-json\/wp\/v2\/users\/62"}],"replies":[{"embeddable":true,"href":"https:\/\/learnsteer.sasnaka.org\/science\/wp-json\/wp\/v2\/comments?post=12483"}],"version-history":[{"count":46,"href":"https:\/\/learnsteer.sasnaka.org\/science\/wp-json\/wp\/v2\/posts\/12483\/revisions"}],"predecessor-version":[{"id":34942,"href":"https:\/\/learnsteer.sasnaka.org\/science\/wp-json\/wp\/v2\/posts\/12483\/revisions\/34942"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/learnsteer.sasnaka.org\/science\/wp-json\/wp\/v2\/media\/16553"}],"wp:attachment":[{"href":"https:\/\/learnsteer.sasnaka.org\/science\/wp-json\/wp\/v2\/media?parent=12483"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/learnsteer.sasnaka.org\/science\/wp-json\/wp\/v2\/categories?post=12483"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/learnsteer.sasnaka.org\/science\/wp-json\/wp\/v2\/tags?post=12483"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}