{"id":4076,"date":"2021-08-15T22:02:36","date_gmt":"2021-08-15T22:02:36","guid":{"rendered":"https:\/\/mathemerize.com\/?p=4076"},"modified":"2021-11-17T17:45:56","modified_gmt":"2021-11-17T12:15:56","slug":"what-is-agp","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/what-is-agp\/","title":{"rendered":"What is AGP – How to Solve AGP Series"},"content":{"rendered":"

Here you will learn what is agp (Arithmetico Geometric Series) and how to solve agp series.<\/p>\n

Let’s begin –<\/p>\n

What is AGP (Arithmetico – Geometric series)<\/h2>\n

A series, each term of which is formed by multiplying the corresponding term of an A.P. & G.P. is called the Arithmetico-Geometric Series, e.g. 1 + 3x + 5\\(x^2\\) + 7\\(x^3\\) + ……..<\/p>\n

Here 1, 3, 5, ……. are in A.P. & 1, x, \\(x^2\\), \\(x^3\\) …….. are in G.P.<\/p>\n

(a) Sum of N terms of an arithmetico-geometric series<\/strong> :<\/h3>\n

Let \\(S_n\\) = a + (a + d)r + …….. + [a + (n-1)d]\\(r^{n-1}\\)<\/p>\n

then \\(S_n\\) = \\(a\\over {1-r}\\) + \\(dr({1}-{r}^{n-1})\\over (1-r)^2\\) – \\([a + (n-1)d]r^2\\over {1-r}\\), r \\(\\ne\\) 1<\/p>\n

(b) Sum of Infinity<\/strong> :<\/h3>\n

If 0 < |r| < 1 & n \\(\\rightarrow\\) \\(\\infty\\)\u00a0 \u00a0\\(\\displaystyle \\lim_{x \\to \\infty}\\) \\(r^n\\) = 0, \\(S_{\\infty}\\) = \\(a\\over {1-r}\\) + \\(dr\\over (1-r)^2\\)<\/p>\n\n\n

Example : <\/span>Find the sum of series 4 – 9x + 16\\(x^2\\) – 25\\(x^3\\) + 36\\(x^4\\) – 49\\(x^5\\) + \n\t\t ……. \\(\\infty\\)<\/p>\n

Solution : <\/span>Let S = 4 – 9x + 16\\(x^2\\) – 25\\(x^3\\) + 36\\(x^4\\) – 49\\(x^5\\) + ……. \\(\\infty\\)

\n\t\t -Sx = -4x + 9\\(x^2\\) – 16\\(x^3\\) + 25\\(x^4\\) – 36\\(x^5\\) + ……. \\(\\infty\\)

\n\t\t On Subtraction, we get

\n\t\t S(1 + x) = 4 – 5x + 7\\(x^2\\) – 9\\(x^3\\) + 11\\(x^4\\) – 13\\(x^5\\) + ……. \\(\\infty\\)

\n\t\t -S(1 + x)x = -4x + 5\\(x^2\\) – 7\\(x^3\\) + 9\\(x^4\\) – 11\\(x^5\\) + ……. \\(\\infty\\)

\n\t\t On Subtraction, we get

\n\t\t S\\((1 + x)^2\\) = 4 – x + 2\\(x^2\\) – 2\\(x^3\\) + 2\\(x^4\\) – 2\\(x^5\\) + ……. \\(\\infty\\)

\n\t\t = 4 – x + 2\\(x^2\\)(1 – x + \\(x^2\\) – \\(x^3\\) + ……. \\(\\infty\\)) = 4 – x + \\(2x^2\\over {1+x}\\)\n\t\t = \\({4 + 3x + x^2}\\over {1+x}\\)

\n\t\t S = \\({4 + 3x + x^2}\\over (1+x)^3\\)

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

RESULTS<\/h4>\n

(a)  \\({\\sum}_{r=1}^{n\u200e}\\)r = \\(n(n + 1)\\over 2\\)    (sum of the first n natural numbers)<\/p>\n

(b)  \\({\\sum}_{r=1}^{n\u200e} r^2\\) = \\(n(n + 1)(2n + 1)\\over 6\\)    (sum of the squares of the first n natural numbers)<\/p>\n

(c)  \\({\\sum}_{r=1}^{n\u200e} r^3\\) = \\(n^2(n + 1)^2\\over 4\\)    (sum of the cubes of the first n natural numbers)<\/p>\n

(d)  \\({\\sum}_{r=1}^{n\u200e} r^4\\) = \\(n(n + 1)(2n + 1)(3n^2 + 3n -1)\\over 30\\)<\/p>\n

(e)  \\({\\sum}_{r=1}^{n\u200e}\\)(2r – 1) = \\(n^2\\)    (sum of the first n odd natural numbers)<\/p>\n

(f)  \\({\\sum}_{r=1}^{n\u200e}\\)2r = n(n + 1)    (sum of the first n even natural numbers)<\/p>\n

Note :<\/strong><\/p>\n

If \\(n^{th}\\) terms of a sequence is given by \\(T_n\\) = \\(an^3\\) + \\(bn^2\\) + cn + d where a, b, c, d are constants,<\/p>\n

then sum of n terms \\(S_n\\) = \\(\\sum T_n\\) = a\\(\\sum n^3\\) + b\\(\\sum n^2\\) + c\\(\\sum n\\) + \\(\\sum d\\)<\/p>\n

This can be evaluated using the above results.<\/p>\n\n\n

Example : <\/span> Sum upto 16 terms of the series \\(1^3\\over 1\\) + \\(1^3 + 2^3\\over 1 + 3\\) + \n\t\t \\(1^3 + 2^3 + 3^3\\over 1 + 3 + 5\\) +…….. is<\/p>\n

Solution : <\/span>\\(t_n\\) = \\(1^3 + 2^3 + 3^3 + …. + n^3\\over 1 + 3 + 5 + ….. + (2n – 1)\\)

\n\t\t       = \\({n^2(n + 1)^2\\over 4}\\over n\/2{[2+2(n – 1)]}\\) = \\({n^2(n + 1)^2\\over 4}\\over n^2\\)\n\t\t = \\({(n + 1)^2\\over 4}\\) = \\(n^2\\over 4\\) + \\(n\\over 4\\) + \\(1\\over 4\\)

\n\t\t \\(\\therefore\\)    \\(S_n\\) = \\(\\sum t_n\\) + \\(1\\over 4\\)\\(\\sum n^2\\) + \\(1\\over 2\\)\\(\\sum n\\) + \\(1\\over 4\\)\\(\\sum 1\\)\n\t\t = \\(1\\over 4\\).\\(n(n + 1)(2n + 1)\\over 6\\) + \\(1\\over 2\\).\\(n(n + 1)\\over 2\\) + \\(1\\over 4\\).n

\n\t\t \\(\\therefore\\)    \\(S_{16}\\) = \\(16.17.33\\over 24\\) + \\(16.17\\over 4\\) + \\(16\\over 4\\) = 446.

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

\n
Next – Formula for Geometric Progression (GP)<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Relation Between Arithmetic Geometric and Harmonic mean<\/a><\/div>\n<\/div>\n\n\n\n

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

Here you will learn what is agp (Arithmetico Geometric Series) and how to solve agp series. Let’s begin – What is AGP (Arithmetico – Geometric series) A series, each term of which is formed by multiplying the corresponding term of an A.P. & G.P. is called the Arithmetico-Geometric Series, e.g. 1 + 3x + 5\\(x^2\\) …<\/p>\n

What is AGP – How to Solve AGP Series<\/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":[153,154,156,155],"yoast_head":"\nWhat is AGP - How to Solve AGP Series - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn what is agp (Arithmetico Geometric Series) and how to solve agp series.\" \/>\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\/what-is-agp\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"What is AGP - How to Solve AGP Series - Mathemerize\" \/>\n<meta property=\"og:description\" content=\"In this post you will learn what is agp (Arithmetico Geometric Series) and how to solve agp series.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathemerize.com\/what-is-agp\/\" \/>\n<meta property=\"og:site_name\" content=\"Mathemerize\" \/>\n<meta property=\"article:published_time\" content=\"2021-08-15T22:02:36+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2021-11-17T12:15:56+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=\"3 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathemerize.com\/what-is-agp\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/what-is-agp\/\"},\"author\":{\"name\":\"mathemerize\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df\"},\"headline\":\"What is AGP – How to Solve AGP Series\",\"datePublished\":\"2021-08-15T22:02:36+00:00\",\"dateModified\":\"2021-11-17T12:15:56+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathemerize.com\/what-is-agp\/\"},\"wordCount\":504,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathemerize.com\/#organization\"},\"keywords\":[\"agp series\",\"agp series formula\",\"agp series sum\",\"sum of infinite agp series formula\"],\"articleSection\":[\"Sequences & Series\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathemerize.com\/what-is-agp\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathemerize.com\/what-is-agp\/\",\"url\":\"https:\/\/mathemerize.com\/what-is-agp\/\",\"name\":\"What is AGP - How to Solve AGP Series - Mathemerize\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/#website\"},\"datePublished\":\"2021-08-15T22:02:36+00:00\",\"dateModified\":\"2021-11-17T12:15:56+00:00\",\"description\":\"In this post you will learn what is agp (Arithmetico Geometric Series) and how to solve agp series.\",\"breadcrumb\":{\"@id\":\"https:\/\/mathemerize.com\/what-is-agp\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/mathemerize.com\/what-is-agp\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/mathemerize.com\/what-is-agp\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/mathemerize.com\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"What is AGP – How to Solve AGP Series\"}]},{\"@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":"What is AGP - How to Solve AGP Series - Mathemerize","description":"In this post you will learn what is agp (Arithmetico Geometric Series) and how to solve agp series.","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\/what-is-agp\/","og_locale":"en_US","og_type":"article","og_title":"What is AGP - How to Solve AGP Series - Mathemerize","og_description":"In this post you will learn what is agp (Arithmetico Geometric Series) and how to solve agp series.","og_url":"https:\/\/mathemerize.com\/what-is-agp\/","og_site_name":"Mathemerize","article_published_time":"2021-08-15T22:02:36+00:00","article_modified_time":"2021-11-17T12:15:56+00:00","author":"mathemerize","twitter_card":"summary_large_image","twitter_misc":{"Written by":"mathemerize","Est. reading time":"3 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/mathemerize.com\/what-is-agp\/#article","isPartOf":{"@id":"https:\/\/mathemerize.com\/what-is-agp\/"},"author":{"name":"mathemerize","@id":"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df"},"headline":"What is AGP – How to Solve AGP Series","datePublished":"2021-08-15T22:02:36+00:00","dateModified":"2021-11-17T12:15:56+00:00","mainEntityOfPage":{"@id":"https:\/\/mathemerize.com\/what-is-agp\/"},"wordCount":504,"commentCount":0,"publisher":{"@id":"https:\/\/mathemerize.com\/#organization"},"keywords":["agp series","agp series formula","agp series sum","sum of infinite agp series formula"],"articleSection":["Sequences & Series"],"inLanguage":"en-US","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/mathemerize.com\/what-is-agp\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/mathemerize.com\/what-is-agp\/","url":"https:\/\/mathemerize.com\/what-is-agp\/","name":"What is AGP - How to Solve AGP Series - Mathemerize","isPartOf":{"@id":"https:\/\/mathemerize.com\/#website"},"datePublished":"2021-08-15T22:02:36+00:00","dateModified":"2021-11-17T12:15:56+00:00","description":"In this post you will learn what is agp (Arithmetico Geometric Series) and how to solve agp series.","breadcrumb":{"@id":"https:\/\/mathemerize.com\/what-is-agp\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/mathemerize.com\/what-is-agp\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/mathemerize.com\/what-is-agp\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/mathemerize.com\/"},{"@type":"ListItem","position":2,"name":"What is AGP – How to Solve AGP Series"}]},{"@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\/4076"}],"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=4076"}],"version-history":[{"count":8,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts\/4076\/revisions"}],"predecessor-version":[{"id":8251,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts\/4076\/revisions\/8251"}],"wp:attachment":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/media?parent=4076"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/categories?post=4076"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/tags?post=4076"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}