{"id":6475,"date":"2021-10-16T17:34:15","date_gmt":"2021-10-16T12:04:15","guid":{"rendered":"https:\/\/mathemerize.com\/?p=6475"},"modified":"2021-11-18T22:37:21","modified_gmt":"2021-11-18T17:07:21","slug":"middle-term-in-binomial-expansion","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/middle-term-in-binomial-expansion\/","title":{"rendered":"Middle Term in Binomial Expansion"},"content":{"rendered":"

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

Let’s begin – <\/p>\n

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

Since the binomial expansion of \\((x + a)^n\\) contains (n + 1) terms. Therefore,<\/p>\n

\n

(1) If n is even, then \\({n\\over 2} + 1\\) th term is the middle term.<\/p>\n

(2) If n is odd, then \\({n + 1\\over 2}\\) th and \\({n + 3\\over 2}\\) th terms are the two middle terms.<\/p>\n<\/blockquote>\n

Example<\/strong><\/span> : Find the middle term in the expansion of \\(({2\\over 3}x^2 – {3\\over 2x})^{20}\\).<\/p>\n

Solution<\/span><\/strong> : Here, n = 20, which is an even number.<\/p>\n

So, \\({20\\over 2} + 1\\)th term i.e. 11th term is the middle term.<\/p>\n

Hence, the middle term = \\(T_{11}\\)<\/p>\n

\\(T_{11}\\) = \\(T_{10 + 1}\\) = \\(^{20}C_{10}\\) \\(({2\\over 3}x^2)^{20 – 10}\\) \\((-{3\\over 2x})^{10}\\)<\/p>\n

= \\(^{20}C_{10} x^{10}\\)<\/p>\n

Example<\/span><\/strong> : Find the middle term in the expansion of \\((3x – {x^3\\over 6})^7\\).<\/p>\n

Solution<\/span><\/strong> : The given expression is \\((3x – {x^3\\over 6})^7\\).<\/p>\n

Here n = 7, which is an odd number.<\/p>\n

So, \\(({7 + 1\\over 2}\\)) th and \\(({7 + 1\\over 2} + 1)\\) th i.e.  4th and 5th terms are two middle terms.<\/p>\n

Now, \\(T_{4}\\) = \\(T_{3 + 1}\\) = \\(^{7}C_{3}\\) \\((3x)^{7 – 3}\\) \\((-{x^3\\over 6})^{3}\\)<\/p>\n

\\(\\implies\\) \\(T_{4}\\) = \\((-1)^3\\) \\(^{7}C_{3}\\) \\((3x)^{4}\\) \\(({x^3\\over 6})^{3}\\)<\/p>\n

\\(\\implies\\) \\(T_{4}\\) = -35 \\(\\times\\) 81 \\(x^4\\) \\(\\times\\) \\(x^9\\over 216\\) = -\\(105x^{13}\\over 8\\)<\/p>\n

and,  \\(T_{5}\\) = \\(T_{5 + 1}\\) = \\(^{7}C_{4}\\) \\((3x)^{7 – 4}\\) \\((-{x^3\\over 6})^{4}\\)<\/p>\n

\\(\\implies\\) \\(T_{5}\\) = \\(^{7}C_{4}\\) \\((3x)^{3}\\) \\(({x^3\\over 6})^{4}\\)<\/p>\n

\\(\\implies\\) \\(T_{5}\\)= 35 \\(\\times\\) 27 \\(x^3\\) \\(\\times\\) \\(x^{12}\\over 1296\\) = \\(35x^{15}\\over 48\\)<\/p>\n

Hence, the middle terms are -\\(105x^{13}\\over 8\\) and \\(35x^{15}\\over 48\\).<\/p>\n\n\n

\n
Previous – General Term in Binomial Expansion<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn formula to find middle term in binomial expansion with examples. Let’s begin –  Middle Term in Binomial Expansion Since the binomial expansion of \\((x + a)^n\\) contains (n + 1) terms. Therefore, (1) If n is even, then \\({n\\over 2} + 1\\) th term is the middle term. (2) If n …<\/p>\n

Middle 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":[173,172,171],"yoast_head":"\nMiddle Term in Binomial Expansion - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn formula to find middle 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\/middle-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=\"Middle Term in Binomial Expansion - 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