{"id":4015,"date":"2021-08-13T20:51:48","date_gmt":"2021-08-13T20:51:48","guid":{"rendered":"https:\/\/mathemerize.com\/?p=4015"},"modified":"2021-11-17T17:55:03","modified_gmt":"2021-11-17T12:25:03","slug":"formula-for-sum-of-ap-series","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/formula-for-sum-of-ap-series\/","title":{"rendered":"Formula for Sum of AP Series | Properties of AP"},"content":{"rendered":"

Here, you will learn what is arithmetic progression (ap) and formula for sum of ap series and properties of ap.<\/p>\n

Let’s begin –<\/p>\n

The indicated sum of the terms of a sequence. In the case of a finite sequence \\(a_1\\), \\(a_2\\), \\(a_3\\),………,\\(a_n\\) the corresponding series is \\(a_1\\) + \\(a_2\\) + \\(a_3\\) + ……… + \\(a_n\\) = \\({\\sum_{k=1}^{n}a_k}\\). This series has a finite or limited numbers of terms and is called a finite series.<\/p>\n

Formula for Sum of AP Series<\/h2>\n

A.P. is a sequence whose terms differ by a fixed number. This fixed number is called the common difference. If a is the first term & d the common difference, then A.P. can be written as<\/p>\n

a, a + d, a + 2d, ………, a + (n – 1)d, ……..<\/p>\n

(a)\u00a0 nth term of AP<\/h3>\n
\n

\\(T_n\\) = a+(n-1)d,<\/p>\n

where d = \\(t_n\\) – \\(t_{n-1}\\)<\/p>\n<\/blockquote>\n

(b)\u00a0 The sum of the first n terms<\/h3>\n
\n

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

\\(S_n\\) = \\(n \\over 2\\)[a + l]\u00a0<\/p>\n

where l is \\(n^{th}\\) term.<\/p>\n<\/blockquote>\n

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

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

(i)\u00a0 \\(n^{th}\\) term of an A.P. is of the form An + B i.e. a linear expression in ‘n’, in such a case the coefficient of n is the common difference of the A.P. i.e. A.<\/p>\n

(ii)\u00a0 Sum of first ‘n’ terms of an A.P. is of the form \\(An^2\\) + Bn i.e. a quadratic expression in ‘n’, in such case the common difference is twice the cofficient of \\(n^2\\). i.e. 2A<\/p>\n

(iii)\u00a0 Also \\(n^{th}\\) term \\(T_n\\) = \\(S_n\\) – \\(S_{n-1}\\)<\/p>\n\n\n

Example : <\/span>If (x + 1), 3x and (4x + 2) are first three terms of an A.P. then its \\(5^{th}\\) term is-<\/p>\n

Solution : <\/span>(x + 1), 3x, (4x + 2) are in AP
\n\t\t => 3x – (x + 1) = (4x + 2) – 3x   =>  x = 3
\n\t\t => a = 4, d = 9 – 4 = 5

\n\t\t => \\(T_5\\) = 4 + (4)5 = 24

\nHence, its \\(5^{th}\\) term is 24

<\/p>\n\n\n

Properties of AP<\/h2>\n

(a)  If each term of an A.P. is increased, decreased, multiplied or divided by the some nonzero number, then the resulting sequence is also an A.P.<\/p>\n

(b)  <\/p>\n

Three numbers in A.P. : a-d, a, a+d<\/strong><\/p>\n

Four numbers in A.P. : a-3d, a-d, a+d, a+3d<\/strong><\/p>\n

Five numbers in A.P. : a-2d, a-d, a, a+d, a+2d<\/strong><\/p>\n

Six numbers in A.P. : a-5d, a-3d, a-d, a+d, a+3d, a+5d<\/strong> etc.<\/p>\n

(c)  The common difference can be zero, positive or negative.<\/p>\n

(d)  \\(k^{th}\\) term from the last = (n – k+1)th term from the beginning (If total number of terms = n).<\/p>\n

(e)  The sum of the two terms of an AP equidistant from the beginning & end is constant and equal to the sum of first & last terms. => \\(T_k\\) + \\(T_{n-k+1}\\) = constant = a + l.<\/p>\n

(f)  Any term of an AP (except the first) is equal to half the sum of terms which are equidistant from it.  \\(a_n\\) = (1\/2)(\\(a_{n-k}\\) + \\(a_{n+k}\\)), k < n  <\/p>\n

(g)  If a, b, c are in AP, then 2b = a + c. <\/p>\n\n\n

Example : <\/span> Four numbers are in A.P. If their sum is 20 and the sum of their squares is 120, then the middle terms are-<\/p>\n

Solution : <\/span>Let the numbers are a-3d, a-d, a+d, a+3d

given, a-3d + a-d + a+d + a+3d = 20\n\t\t   => 4a = 20   => a=5

and \\((a-3d)^2\\) + \\((a-d)^2\\) + \\((a+d)^2\\) + \\((a+3d)^2\\) = 120

\n\t\t => 4\\(a^2\\) + 20\\(d^2\\) = 120

\n\t\t => 4 x \\(5^2\\) + 20\\(d^2\\) = 20    =>    \\(d^2\\) = 1    =>    d = \\(\\pm\\)1

\n\t\t Hence numbers are 2,4,6,8    or    8,4,6,2

<\/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

Prove that the sum of first n natural numbers is \\(n(n+1)\\over 2\\)<\/a><\/p>\n\n\n

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

Here, you will learn what is arithmetic progression (ap) and formula for sum of ap series and properties of ap. Let’s begin – The indicated sum of the terms of a sequence. In the case of a finite sequence \\(a_1\\), \\(a_2\\), \\(a_3\\),………,\\(a_n\\) the corresponding series is \\(a_1\\) + \\(a_2\\) + \\(a_3\\) + ……… + \\(a_n\\) …<\/p>\n

Formula for Sum of AP Series | Properties of 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":[145,144,151,148,147],"yoast_head":"\nFormula for Sum of AP Series | Properties of AP - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn what is arithmetic progression (ap) and formula for sum of ap series and properties of ap.\" \/>\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-sum-of-ap-series\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Formula for Sum of AP Series | Properties of AP - 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