{"id":6473,"date":"2021-10-16T17:33:02","date_gmt":"2021-10-16T12:03:02","guid":{"rendered":"https:\/\/mathemerize.com\/?p=6473"},"modified":"2021-11-20T01:34:26","modified_gmt":"2021-11-19T20:04:26","slug":"general-term-in-binomial-expansion","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/general-term-in-binomial-expansion\/","title":{"rendered":"General Term in Binomial Expansion"},"content":{"rendered":"

Here you will learn formula to find the general term in binomial expansion with examples.<\/p>\n

Let’s begin –<\/p>\n

General Term in Binomial Expansion<\/h2>\n

We have,<\/p>\n

\n

\\((x + a)^n\\) = \\(^{n}C_0 x^n a^0\\) + \\(^{n}C_1 x^{n – 1} a^1\\) + …………… + \\(^{n}C_r x^{n – r} a^r\\) + …………… + \\(^{n}C_n x^0 a^n\\)<\/p>\n<\/blockquote>\n

We find that : The first term = \\(^{n}C_0 x^n a^0\\)<\/p>\n

The second term = \\(^{n}C_1 x^{n – 1} a^1\\)<\/p>\n

The third term = \\(^{n}C_2 x^{n – 2} a^2\\)<\/p>\n

The fourth term = \\(^{n}C_3 x^{n – 3} a^3\\),\u00a0 and so on.<\/p>\n

We thus observe that the suffix of C in any term is one less than the number of terms, the index of x is n minus the suffix of C and the index of a is the same as the suffix of C.<\/p>\n

Hence, the (r + 1)th term is given by \\(^{n}C_r x^{n – r} a^r\\)<\/p>\n

Thus, if \\(T_{r + 1}\\) denotes the (r + 1)th term, then<\/p>\n

General Term :<\/strong><\/h4>\n
\n

\\(T_{r + 1}\\) = \\(^{n}C_r x^{n – r} a^r\\)<\/p>\n<\/blockquote>\n

This is called the general term<\/strong>, because by giving different values to r we can determine all terms of the expansion.<\/p>\n

In the binomial expansion of \\((x – a)^n\\), the general term is given by<\/p>\n

\n

\\(T_{r + 1}\\) = \\((-1)^r\\)\\(^{n}C_r x^{n – r} a^r\\)<\/p>\n<\/blockquote>\n

In the binomial expansion of \\((1 + x)^n\\), we have<\/p>\n

\n

\\(T_{r + 1}\\) = \\(^nC_r x^r\\)<\/p>\n<\/blockquote>\n

In the binomial expansion of \\((1 – x)^n\\), we have<\/p>\n

\n

\\(T_{r + 1}\\) = \\((-1)^r\\)\\(^nC_r x^r\\)<\/p>\n<\/blockquote>\n

Nth term from the End :<\/strong><\/h4>\n
\n

In the binomial expansion of \\((x + a)^n\\), the rth term from the end is ((n + 1) – r + 1) = (n – r + 2)th term form the beginning.<\/p>\n<\/blockquote>\n

Example<\/span><\/strong> : Write the general term in the expansion of \\((x^2 – y)^6\\).<\/p>\n

Solution<\/span><\/strong> : We have, \\((x^2 – y)^6\\) = \\(|(x^2 + (-y)|^6\\)<\/p>\n

The general term in the expansion of the above binomial is given by\u00a0<\/p>\n

\\(T_{r + 1}\\) = \\(^{n}C_r x^{n – r} a^r\\)<\/p>\n

\\(\\implies\\) \\(T_{r + 1}\\) = \\(^{6}C_r (x^2)^{6 – r} (-y)^r\\)<\/p>\n

\\(\\implies\\) \\(T_{r + 1}\\) = \\((-1)^r\\)\\(^{6}C_r x^{12 – 2r} y^r\\)<\/p>\n

\n

Related Questions<\/h3>\n

Find the 9th term in the expansion of \\(({x\\over a} \u2013 {3a\\over x^2})^{12}\\).<\/a><\/p>\n

Find the 10th term in the binomial expansion of \\((2x^2 + {1\\over x})^{12}\\).<\/a><\/p>\n\n\n

\n
Next – Middle Term in Binomial Expansion<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Formula for Binomial Theorem<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn formula to find the general term in binomial expansion with examples. Let’s begin – General Term in Binomial Expansion We have, \\((x + a)^n\\) = \\(^{n}C_0 x^n a^0\\) + \\(^{n}C_1 x^{n – 1} a^1\\) + …………… + \\(^{n}C_r x^{n – r} a^r\\) + …………… + \\(^{n}C_n x^0 a^n\\) We find that …<\/p>\n

General Term in Binomial Expansion<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"default","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[40],"tags":[173,172,175],"yoast_head":"\nGeneral Term in Binomial Expansion - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn formula to find the general term in binomial expansion with examples.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathemerize.com\/general-term-in-binomial-expansion\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"General Term in Binomial Expansion - Mathemerize\" \/>\n<meta property=\"og:description\" content=\"In this post you will learn formula to find the general term in binomial expansion with examples.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathemerize.com\/general-term-in-binomial-expansion\/\" \/>\n<meta property=\"og:site_name\" content=\"Mathemerize\" \/>\n<meta property=\"article:published_time\" content=\"2021-10-16T12:03:02+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2021-11-19T20:04:26+00:00\" \/>\n<meta name=\"author\" content=\"mathemerize\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"mathemerize\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"2 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathemerize.com\/general-term-in-binomial-expansion\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/general-term-in-binomial-expansion\/\"},\"author\":{\"name\":\"mathemerize\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df\"},\"headline\":\"General Term in Binomial Expansion\",\"datePublished\":\"2021-10-16T12:03:02+00:00\",\"dateModified\":\"2021-11-19T20:04:26+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathemerize.com\/general-term-in-binomial-expansion\/\"},\"wordCount\":403,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathemerize.com\/#organization\"},\"keywords\":[\"binomial expansion\",\"binomial theorem\",\"General Term in Binomial Expansion\"],\"articleSection\":[\"Binomial Theorem\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathemerize.com\/general-term-in-binomial-expansion\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathemerize.com\/general-term-in-binomial-expansion\/\",\"url\":\"https:\/\/mathemerize.com\/general-term-in-binomial-expansion\/\",\"name\":\"General Term in Binomial Expansion - Mathemerize\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/#website\"},\"datePublished\":\"2021-10-16T12:03:02+00:00\",\"dateModified\":\"2021-11-19T20:04:26+00:00\",\"description\":\"In this post you will learn formula to find the general term in binomial expansion with examples.\",\"breadcrumb\":{\"@id\":\"https:\/\/mathemerize.com\/general-term-in-binomial-expansion\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/mathemerize.com\/general-term-in-binomial-expansion\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/mathemerize.com\/general-term-in-binomial-expansion\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/mathemerize.com\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"General Term in Binomial Expansion\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/mathemerize.com\/#website\",\"url\":\"https:\/\/mathemerize.com\/\",\"name\":\"Mathemerize\",\"description\":\"Maths Tutorials - Study Math Online\",\"publisher\":{\"@id\":\"https:\/\/mathemerize.com\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/mathemerize.com\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"en-US\"},{\"@type\":\"Organization\",\"@id\":\"https:\/\/mathemerize.com\/#organization\",\"name\":\"Mathemerize\",\"url\":\"https:\/\/mathemerize.com\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/logo\/image\/\",\"url\":\"https:\/\/i1.wp.com\/mathemerize.com\/wp-content\/uploads\/2021\/05\/logo.png?fit=140%2C96&ssl=1\",\"contentUrl\":\"https:\/\/i1.wp.com\/mathemerize.com\/wp-content\/uploads\/2021\/05\/logo.png?fit=140%2C96&ssl=1\",\"width\":140,\"height\":96,\"caption\":\"Mathemerize\"},\"image\":{\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/logo\/image\/\"},\"sameAs\":[\"https:\/\/www.instagram.com\/mathemerize\/\"]},{\"@type\":\"Person\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df\",\"name\":\"mathemerize\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/f0649d8b9c9f4ba7f1682b12d040d2a3?s=96&d=mm&r=g\",\"contentUrl\":\"https:\/\/secure.gravatar.com\/avatar\/f0649d8b9c9f4ba7f1682b12d040d2a3?s=96&d=mm&r=g\",\"caption\":\"mathemerize\"},\"sameAs\":[\"https:\/\/mathemerize.com\"],\"url\":\"https:\/\/mathemerize.com\/author\/mathemerize\/\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"General Term in Binomial Expansion - Mathemerize","description":"In this post you will learn formula to find the general term in binomial expansion with examples.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/mathemerize.com\/general-term-in-binomial-expansion\/","og_locale":"en_US","og_type":"article","og_title":"General Term in Binomial Expansion - Mathemerize","og_description":"In this post you will learn formula to find the general term in binomial expansion with examples.","og_url":"https:\/\/mathemerize.com\/general-term-in-binomial-expansion\/","og_site_name":"Mathemerize","article_published_time":"2021-10-16T12:03:02+00:00","article_modified_time":"2021-11-19T20:04:26+00:00","author":"mathemerize","twitter_card":"summary_large_image","twitter_misc":{"Written by":"mathemerize","Est. reading time":"2 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/mathemerize.com\/general-term-in-binomial-expansion\/#article","isPartOf":{"@id":"https:\/\/mathemerize.com\/general-term-in-binomial-expansion\/"},"author":{"name":"mathemerize","@id":"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df"},"headline":"General Term in Binomial Expansion","datePublished":"2021-10-16T12:03:02+00:00","dateModified":"2021-11-19T20:04:26+00:00","mainEntityOfPage":{"@id":"https:\/\/mathemerize.com\/general-term-in-binomial-expansion\/"},"wordCount":403,"commentCount":0,"publisher":{"@id":"https:\/\/mathemerize.com\/#organization"},"keywords":["binomial expansion","binomial theorem","General Term in Binomial Expansion"],"articleSection":["Binomial Theorem"],"inLanguage":"en-US","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/mathemerize.com\/general-term-in-binomial-expansion\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/mathemerize.com\/general-term-in-binomial-expansion\/","url":"https:\/\/mathemerize.com\/general-term-in-binomial-expansion\/","name":"General Term in Binomial Expansion - Mathemerize","isPartOf":{"@id":"https:\/\/mathemerize.com\/#website"},"datePublished":"2021-10-16T12:03:02+00:00","dateModified":"2021-11-19T20:04:26+00:00","description":"In this post you will learn formula to find the general term in binomial expansion with examples.","breadcrumb":{"@id":"https:\/\/mathemerize.com\/general-term-in-binomial-expansion\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/mathemerize.com\/general-term-in-binomial-expansion\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/mathemerize.com\/general-term-in-binomial-expansion\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/mathemerize.com\/"},{"@type":"ListItem","position":2,"name":"General Term in Binomial Expansion"}]},{"@type":"WebSite","@id":"https:\/\/mathemerize.com\/#website","url":"https:\/\/mathemerize.com\/","name":"Mathemerize","description":"Maths Tutorials - Study Math Online","publisher":{"@id":"https:\/\/mathemerize.com\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/mathemerize.com\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"en-US"},{"@type":"Organization","@id":"https:\/\/mathemerize.com\/#organization","name":"Mathemerize","url":"https:\/\/mathemerize.com\/","logo":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/mathemerize.com\/#\/schema\/logo\/image\/","url":"https:\/\/i1.wp.com\/mathemerize.com\/wp-content\/uploads\/2021\/05\/logo.png?fit=140%2C96&ssl=1","contentUrl":"https:\/\/i1.wp.com\/mathemerize.com\/wp-content\/uploads\/2021\/05\/logo.png?fit=140%2C96&ssl=1","width":140,"height":96,"caption":"Mathemerize"},"image":{"@id":"https:\/\/mathemerize.com\/#\/schema\/logo\/image\/"},"sameAs":["https:\/\/www.instagram.com\/mathemerize\/"]},{"@type":"Person","@id":"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df","name":"mathemerize","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/mathemerize.com\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/f0649d8b9c9f4ba7f1682b12d040d2a3?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/f0649d8b9c9f4ba7f1682b12d040d2a3?s=96&d=mm&r=g","caption":"mathemerize"},"sameAs":["https:\/\/mathemerize.com"],"url":"https:\/\/mathemerize.com\/author\/mathemerize\/"}]}},"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts\/6473"}],"collection":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/comments?post=6473"}],"version-history":[{"count":10,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts\/6473\/revisions"}],"predecessor-version":[{"id":8299,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts\/6473\/revisions\/8299"}],"wp:attachment":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/media?parent=6473"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/categories?post=6473"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/tags?post=6473"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}