{"id":4766,"date":"2021-08-25T15:22:52","date_gmt":"2021-08-25T15:22:52","guid":{"rendered":"https:\/\/mathemerize.com\/?p=4766"},"modified":"2022-01-16T16:57:35","modified_gmt":"2022-01-16T11:27:35","slug":"sum-of-gp-to-infinity","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/sum-of-gp-to-infinity\/","title":{"rendered":"Sum of GP to Infinity – Example and Proof"},"content":{"rendered":"

Here you will learn sum of gp to infinity (sum of infinite gp) and its proof with examples.<\/p>\n

Let’s begin –<\/p>\n

Sum of GP to Infinity (Sum of Infinite GP)<\/h2>\n

The sum of an infinite GP with first term a and common ratio r(-1 < r < 1 i.e. ,  | r | < 1) is<\/p>\n

\n

S = \\(a\\over 1-r\\)<\/p>\n<\/blockquote>\n

Also Read<\/strong> : Sum of GP Series Formula | Properties of GP<\/a><\/p>\n

Proof : <\/strong>Consider an infinite GP with first term a and common ratio r, where -1 < r < 1 i.e. , | r | < 1. The sum of n terms of this GP is given by<\/p>\n

\\(S_n\\) = a\\({1-r^n}\\over {1-r}\\) = \\(a\\over {1-r}\\) – \\(ar^n\\over {1-r}\\)           ……..(i)<\/p>\n

Since -1 < r < 1, therefore \\(r^n\\) decreases as n increases and \\(r^n\\) tends to zero as n tends to infinity i.e. \\(r^n\\) \\(\\rightarrow\\) 0 as n \\(\\rightarrow\\) \\(\\infty\\).<\/p>\n

\\(\\therefore\\)    \\(ar^n\\over {1-r}\\) \\(\\rightarrow\\) 0 as n \\(\\rightarrow\\) \\(\\infty\\).<\/p>\n

Hence from (i), the sum of an infinite GP is given by<\/p>\n

S = \\(lim_{n \\to \\infty}\\) \\(S_n\\) = \\(lim_{n \\to \\infty}\\) ( \\(a\\over {1-r}\\) – \\(ar^n\\over {1-r}\\) ) = \\(a\\over 1-r\\), if | r | < 1<\/p>\n

Note : <\/strong>If r \\(\\ge\\) 1, then the sum of an infinite GP tends to infinity.<\/p>\n\n\n

Example : <\/span>Find the sum to infinity of the GP \\(-5\\over 4\\), \\(5\\over 16\\), \\(-5\\over 64\\), ……<\/p>\n

Solution : <\/span>The given GP has the first term a = -5\/4 and the common ratio r = -1\/4

\nAlso | r | < 1.

\nHence the sum of an infinite GP is given by S = \\(a\\over {1-r}\\)

\nS = \\(-5\/4\\over {1-(-1\/4)}\\) = -1

<\/p>\n\n\n\n

Example : <\/span>The sum of an infinite GP is 57 and the sum of their cubes is 9747, find the GP.<\/p>\n

Solution : <\/span>Let a be the first term and r be the common ratio of the GP. Then

\nSum = 57 \\(\\implies\\) \\(a\\over 1-r\\) = 57 …….(i)

\nSum of the cubes = 9747

\n\\(\\implies\\) \\(a^3\\) + \\(a^3r^3\\) + \\(a^3r^6\\) + ….. = 9747

\n\\(\\implies\\) \\(a^3\\over {1 – r^3}\\) = 9747 ……..(ii)

\nDividing the cube of (i) by (ii), we get

\n\\(a^3\\over {(1-r)}^3\\) . \\((1-r^3)\\over a^3\\) = \\({(57)}^3\\over 9747\\)

\n\\(\\implies\\) \\(1 – r^3\\over {(1 – r)}^3\\) = 19

\n= \\(1+r+r^2\\over {(1-r)}^2\\)

\n= \\(18r^2\\) – 39r + 18 = 0

\n\\(\\implies\\) (3r-2)(6r-9) = 0

\n\\(\\implies\\) r = 2\/3 or r = 3\/2

\nHence r = 2\/3 [ \\(\\because\\) r \\(\\ne\\) 3\/2, because -1 < r < 1 for an infinite GP]

\nPutting r = 2\/3 in equation (i), we get

\n\\(a\\over {(1-(2\/3))}\\) = 57 \\(\\implies\\) a = 19

\nHence, the GP is 19, 38\/3, 76\/9, …….

<\/p>\n\n\n\n

\n
Next – Method of Difference \u2013 Sequences & Series<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Formula for Geometric Progression (GP)<\/a><\/div>\n<\/div>\n\n\n\n

<\/p>\n","protected":false},"excerpt":{"rendered":"

Here you will learn sum of gp to infinity (sum of infinite gp) and its proof with examples. Let’s begin – Sum of GP to Infinity (Sum of Infinite GP) The sum of an infinite GP with first term a and common ratio r(-1 < r < 1 i.e. ,  | r | < 1) …<\/p>\n

Sum of GP to Infinity – Example and Proof<\/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":[15],"tags":[133,132,135,130,136,131],"yoast_head":"\nSum of GP to Infinity - Example and Proof - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn proof of sum of gp to infinity (sum of infinite gp) 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\/sum-of-gp-to-infinity\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Sum of GP to Infinity - 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