{"id":4762,"date":"2021-08-25T15:09:23","date_gmt":"2021-08-25T15:09:23","guid":{"rendered":"https:\/\/mathemerize.com\/?p=4762"},"modified":"2021-11-17T17:47:44","modified_gmt":"2021-11-17T12:17:44","slug":"formula-for-geometric-progression-gp","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/formula-for-geometric-progression-gp\/","title":{"rendered":"Formula for Geometric Progression (GP)"},"content":{"rendered":"

Here you will learn what is geometric progression and formula for geometric progression with examples.<\/p>\n

Let’s begin –<\/p>\n

Geometric Progression (GP)<\/h2>\n

A sequence of non-zero numbers is called a geometric progression (GP) if the ratio of a term and the term preceding to its always a constant quantity.<\/p>\n

The constant ratio is called the common ration of the GP.<\/p>\n

\n

In other words, a sequence \\(a_1\\), \\(a_2\\), \\(a_3\\) …….. \\(a_n\\) , …….. is called a geometric progression if<\/p>\n

\\(a_{n+1}\\over a_n\\) = constant for all n \\(\\in\\) N<\/p>\n<\/blockquote>\n

Example : The sequence 4, 12, 36, 108, is a GP , because<\/p>\n

\\(12\\over 4\\) = \\(36\\over 12\\) = \\(108\\over 36\\) = ….. = 3, which is a constant.<\/p>\n

Clearly, this sequence is a GP with first term 4 and common ratio 3.<\/p>\n

Also Read<\/strong> : Formula for Arithmetic Progression (AP)<\/a><\/p>\n

Formula for Geometric Progression<\/h2>\n

(a) General Term of a GP<\/h3>\n

Let a be the first term and r be the common ratio of a GP. Then its nth term or general term is \\(ar^{n-1}\\)<\/p>\n

\n

i.e\u00a0 \u00a0\\(a_n\\) = \\(ar^{n-1}\\)<\/p>\n<\/blockquote>\n

Note :\u00a0<\/strong>The whole GP can be written as a, ar, \\(ar^2\\), …… , \\(ar^{n-1}\\)\u00a0 or\u00a0 a, ar, \\(ar^2\\), …… , \\(ar^{n-1}\\), …… according as it is finite or infinite.<\/p>\n

(b) nth term from the end of a finite GP<\/h3>\n

(i) The nth term from the end of a finite GP consisting of m terms is \\(ar^{m-n}\\), where a is the first term and r is the common ratio of the GP.<\/p>\n

\n

\\(\\because\\)\u00a0 \u00a0nth term from the end = (m – n + 1)th term from the beginning\u00a0<\/p>\n

\u00a0= \\(ar^{m-n}\\)<\/p>\n<\/blockquote>\n

(ii) and the nth term from the end of a GP with last term l and common ratio r is given by\u00a0<\/p>\n

\n

\\(a_n\\) = l\\(({1\\over r})^{n-1}\\)<\/p>\n<\/blockquote>\n

Clearly when we look at the terms terms of a GP from the last term and move towards the beginning we find that the progression is a GP with the common ration 1\/r.<\/p>\n

\n

So, nth term from the end = l\\(({1\\over r})^{n-1}\\)<\/p>\n<\/blockquote>\n

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

Example : <\/span>Find the 9th term and the general term of the progression \\(1\\over 4\\), \\(-1\\over 2\\), 1, -2, ……<\/p>\n

Solution : <\/span>The given progresion is clearly a gp with the first term a = \\(1\\over 4\\) and common ratio r = -2.

\n\\(\\therefore\\) 9th term = \\(a_9\\) = \\(ar^{(9-1)}\\) = \\(1\\over 4\\)\\((-2)^{8}\\) = 64

\nand, General term = \\(a_n\\) = \\(ar^{(n-1)}\\) = \\(1\\over 4\\) \\((-2)^{n-1}\\) = \\((-1)^{n-1}\\) \\(2{n-3}\\)

<\/p>\n\n\n

\n

Related Questions<\/h3>\n

If the 4th and 9th terms of a G.P. be 54 and 13122 respectively, find the G.P.<\/a><\/p>\n

Three positive integers form an increasing GP. If the middle term in this GP is doubled, then new numbers are in AP. Then the common ratio of GP is<\/a><\/p>\n\n\n

\n
Next – Sum of GP to Infinity \u2013 Example & Proof<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous -What is AGP \u2013 How to Solve AGP Series<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn what is geometric progression and formula for geometric progression with examples. Let’s begin – Geometric Progression (GP) A sequence of non-zero numbers is called a geometric progression (GP) if the ratio of a term and the term preceding to its always a constant quantity. The constant ratio is called the common …<\/p>\n

Formula for Geometric Progression (GP)<\/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":[137,141,133,132,138,135,142,140],"yoast_head":"\nFormula for Geometric Progression (GP) - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn what is geometric progression and formula for geometric progression 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\/formula-for-geometric-progression-gp\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Formula for Geometric Progression (GP) - 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