{"id":14490,"date":"2020-12-11T07:31:09","date_gmt":"2020-12-11T02:01:09","guid":{"rendered":"https:\/\/www.aplustopper.com\/?p=14490"},"modified":"2020-12-11T09:20:31","modified_gmt":"2020-12-11T03:50:31","slug":"binomial-theorem-index","status":"publish","type":"post","link":"https:\/\/www.aplustopper.com\/binomial-theorem-index\/","title":{"rendered":"Binomial Theorem for any Index"},"content":{"rendered":"<h2>Binomial Theorem for any Index<\/h2>\n<p><iframe loading=\"lazy\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/xMziTBR34_M?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/p>\n<h3>Binomial theorem for positive integral index<\/h3>\n<p>The rule by which any power of binomial can be expanded is called the binomial theorem.<br \/>\nIf n is a positive integer and x, y \u2208\u00a0C then<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-130391\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Binomial-Theorem-for-any-Index-1.jpg\" alt=\"Binomial Theorem for any Index 1\" width=\"597\" height=\"257\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Binomial-Theorem-for-any-Index-1.jpg 597w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Binomial-Theorem-for-any-Index-1-300x129.jpg 300w\" sizes=\"(max-width: 597px) 100vw, 597px\" \/><\/p>\n<h3>Binomial theorem for any Index<\/h3>\n<p><strong>Statement :<\/strong><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-130392\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Binomial-Theorem-for-any-Index-2.jpg\" alt=\"Binomial Theorem for any Index 2\" width=\"516\" height=\"118\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Binomial-Theorem-for-any-Index-2.jpg 516w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Binomial-Theorem-for-any-Index-2-300x69.jpg 300w\" sizes=\"(max-width: 516px) 100vw, 516px\" \/><br \/>\nwhen n is a negative integer or a fraction, where , otherwise expansion will not be possible.<br \/>\nIf first term is not 1, then make first term unity in the following way,<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-130393\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Binomial-Theorem-for-any-Index-3.jpg\" alt=\"Binomial Theorem for any Index 3\" width=\"301\" height=\"66\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Binomial-Theorem-for-any-Index-3.jpg 301w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Binomial-Theorem-for-any-Index-3-300x66.jpg 300w\" sizes=\"(max-width: 301px) 100vw, 301px\" \/><\/p>\n<h3><strong>General term :<\/strong><\/h3>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-130394\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Binomial-Theorem-for-any-Index-4.jpg\" alt=\"Binomial Theorem for any Index 4\" width=\"329\" height=\"53\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Binomial-Theorem-for-any-Index-4.jpg 329w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Binomial-Theorem-for-any-Index-4-300x48.jpg 300w\" sizes=\"(max-width: 329px) 100vw, 329px\" \/><\/p>\n<h3>Some important expansions<\/h3>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-130415\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Binomial-Theorem-for-any-Index-5.jpg\" alt=\"Binomial Theorem for any Index 5\" width=\"546\" height=\"728\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Binomial-Theorem-for-any-Index-5.jpg 546w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Binomial-Theorem-for-any-Index-5-225x300.jpg 225w\" sizes=\"(max-width: 546px) 100vw, 546px\" \/><\/p>\n<h3>Problems on approximation by the binomial theorem :<\/h3>\n<p>We have,<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-130416\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Binomial-Theorem-for-any-Index-6.jpg\" alt=\"Binomial Theorem for any Index 6\" width=\"328\" height=\"55\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Binomial-Theorem-for-any-Index-6.jpg 328w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Binomial-Theorem-for-any-Index-6-300x50.jpg 300w\" sizes=\"(max-width: 328px) 100vw, 328px\" \/><br \/>\nIf <em>x<\/em> is small compared with 1, we find that the values of x<sup>2<\/sup>, x<sup>3<\/sup>, x<sup>4<\/sup>, \u2026.. become smaller and smaller.<br \/>\n\u2234 The terms in the above expansion become smaller and smaller. If <em>x<\/em> is very small compared with 1, we might take 1 as a first approximation to the value of (1 + x)<sup>n<\/sup> or (1 + nx) as a second approximation.<\/p>\n<h3>Three \/ Four consecutive terms or Coefficients<\/h3>\n<p>(1) <strong>If consecutive coefficients are given:<\/strong> In this case divide consecutive coefficients pair wise. We get equations and then solve them.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-130417\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Binomial-Theorem-for-any-Index-7.jpg\" alt=\"Binomial Theorem for any Index 7\" width=\"600\" height=\"152\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Binomial-Theorem-for-any-Index-7.jpg 600w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Binomial-Theorem-for-any-Index-7-300x76.jpg 300w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/p>\n<h3>Some important points<\/h3>\n<p><strong>(1) Pascal&#8217;s Triangle<\/strong><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-130418\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Binomial-Theorem-for-any-Index-8.jpg\" alt=\"Binomial Theorem for any Index 8\" width=\"499\" height=\"230\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Binomial-Theorem-for-any-Index-8.jpg 499w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Binomial-Theorem-for-any-Index-8-300x138.jpg 300w\" sizes=\"(max-width: 499px) 100vw, 499px\" \/><br \/>\nPascal&#8217;s triangle gives the direct binomial coefficients.<br \/>\n<em>Example :<\/em> (x + y)<sup>4<\/sup> = x<sup>4<\/sup> + 4x<sup>3<\/sup>y + 6x<sup>2<\/sup>y<sup>2<\/sup> + 4xy<sup>3<\/sup> + y<sup>4<\/sup>.<\/p>\n<p><strong>(2)<\/strong> <strong>Method for finding terms free from radicals or rational terms in the expansion of (a<sup>1\/p<\/sup> + b<sup>1\/q<\/sup>)<sup>N<\/sup> <\/strong><strong>\u2200 a, b \u2208 prime numbers:<\/strong><br \/>\nFind the general term<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-130419\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Binomial-Theorem-for-any-Index-9.jpg\" alt=\"Binomial Theorem for any Index 9\" width=\"344\" height=\"56\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Binomial-Theorem-for-any-Index-9.jpg 344w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Binomial-Theorem-for-any-Index-9-300x49.jpg 300w\" sizes=\"(max-width: 344px) 100vw, 344px\" \/><br \/>\nPutting the values of 0 \u2264 r \u2264 N, when indices of a and b are integers.<br \/>\nNumber of irrational terms = Total terms \u2013 Number of rational terms.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Binomial Theorem for any Index Binomial theorem for positive integral index The rule by which any power of binomial can be expanded is called the binomial theorem. If n is a positive integer and x, y \u2208\u00a0C then Binomial theorem for any Index Statement : when n is a negative integer or a fraction, where [&hellip;]<\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_genesis_hide_title":false,"_genesis_hide_breadcrumbs":false,"_genesis_hide_singular_image":false,"_genesis_hide_footer_widgets":false,"_genesis_custom_body_class":"","_genesis_custom_post_class":"","_genesis_layout":"","footnotes":""},"categories":[5],"tags":[5572,5571,5575,5573,5574],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v17.8 (Yoast SEO v17.8) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Binomial Theorem for any Index - A Plus Topper<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.aplustopper.com\/binomial-theorem-index\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Binomial Theorem for any Index\" \/>\n<meta property=\"og:description\" content=\"Binomial Theorem for any Index Binomial theorem for positive integral index The rule by which any power of binomial can be expanded is called the binomial theorem. 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