{"id":4494,"date":"2021-08-20T05:25:17","date_gmt":"2021-08-20T05:25:17","guid":{"rendered":"https:\/\/mathemerize.com\/?p=4494"},"modified":"2021-11-17T17:47:23","modified_gmt":"2021-11-17T12:17:23","slug":"formula-for-arithmetic-progression","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/formula-for-arithmetic-progression\/","title":{"rendered":"Formula for Arithmetic Progression (AP)"},"content":{"rendered":"

Here you you will learn what is arithmetic progression (AP) and formula for arithmetic progression.<\/p>\n

Let’s begin –<\/p>\n

Arithmetic Progression (AP)<\/h2>\n

A sequence is called an arithmetic progression if the difference of a term and the previous term is always same\u00a0 i.e.\u00a0<\/p>\n

\n

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

The constant difference generally denoted by d is called the common difference.<\/p>\n<\/blockquote>\n

for example<\/span> : 1, 4, 7, 10 ….. is an AP whose first term is 1 and the common difference is equal to 4 – 1 = 3.<\/p>\n

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

Formula for Arithmetic Progression<\/h2>\n

(a) General term of an AP ( nth term of ap)<\/h3>\n

Let a be the first term and d be the common difference of an AP. Then its nth term or general term is a + (n – 1)d\u00a0<\/p>\n

\n

i.e.\u00a0 \u00a0\\(a_n\\) = a + (n – 1)d.<\/p>\n<\/blockquote>\n

(b) nth term of an AP from the end<\/h3>\n

Let a be the first term and d be the common difference of an AP having m terms. Then nth term from the end is \\((m – n + 1)^{th}\\) term from the beginning.<\/p>\n

\n

\\(\\therefore\\)\u00a0 nth term from the end\u00a0 = \\(a_{m-n+1}\\)<\/p>\n

= a + (m-n+1-1)d = a + (m-n)d<\/p>\n<\/blockquote>\n

Also nth term from the end = \\(a_m\\) + (n-1)(-d)<\/p>\n

[\\(\\because\\)\u00a0 \u00a0Taking \\(a_m\\) as the first term and the common difference equal to ‘-d’ ]<\/p>\n

(c) Sum to n terms of an AP<\/h3>\n

The sum \\(S_n\\) of n terms of an AP with first term ‘a’ and common difference ‘d’ is<\/p>\n

\n

\\(S_n\\) = \\(n\\over 2\\) [2a + (n-1)d]<\/p>\n

or,\u00a0 \\(S_n\\) = \\(n\\over 2\\) [a + l] ,\u00a0 where l = last term = a + (n-1)d<\/p>\n<\/blockquote>\n\n\n

Example : <\/span>Show that the sequence 9, 12, 15, 18, ……. is an AP. find its 16th term, general term sum of first 20 terms.<\/p>\n

Solution : <\/span>We have, (12 – 9) = (15 – 12) = (18 – 15) = 3. Therefore, the given sequence is an AP with the common difference 3.

\nfirst term = a = 9

\n\\(\\therefore\\) 16th term = \\(a_{16}\\) = a + (16-1)d

\n\\(\\implies\\) \\(a_{16}\\) = 9 + 15*3 = 54

\n\\(\\because\\) General term = nth term = \\(a_n\\) = a + (n-1)d

\n\\(\\therefore\\) \\(a_n\\) = 9 + (n-1)*3 = 3n + 6

\nNow, sum of first 20 terms = \\(S_{20}\\) = \\(20\\over 2\\) [2*9 + (20-1)3]

\n\\(S_{20}\\) = 10[18 + 19*3]

\n= 750
<\/p>\n\n\n

\n

Related Questions<\/h3>\n

If 100 times the 100th term of an AP with non-zero common difference equal to the 50 times its 50th term, then the 150th term of AP is<\/a><\/p>\n

If x, y and z are in AP and \\(tan^{-1}x\\), \\(tan^{-1}y\\) and \\(tan^{-1}z\\) are also in AP, then<\/a><\/p>\n\n\n

\n
Next – Formula for Sum of AP Series | Properties of AP<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you you will learn what is arithmetic progression (AP) and formula for arithmetic progression. Let’s begin – Arithmetic Progression (AP) A sequence is called an arithmetic progression if the difference of a term and the previous term is always same\u00a0 i.e.\u00a0 \\(a_{n+1}\\) – \\(a_n\\) = constant (=d) for all n \\(\\in\\) N The constant …<\/p>\n

Formula for Arithmetic Progression (AP)<\/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":[149,145,144,143,150,148,147,146],"yoast_head":"\nFormula for Arithmetic Progression (AP) - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn what is arithmetic progression (AP) and formula for arithmetic 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-arithmetic-progression\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Formula for Arithmetic Progression (AP) - 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