{"id":4770,"date":"2021-08-25T15:29:01","date_gmt":"2021-08-25T15:29:01","guid":{"rendered":"https:\/\/mathemerize.com\/?p=4770"},"modified":"2022-01-16T16:57:41","modified_gmt":"2022-01-16T11:27:41","slug":"method-of-difference-sequences-series","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/method-of-difference-sequences-series\/","title":{"rendered":"Method of Difference – Sequences and Series"},"content":{"rendered":"

Here you will learn method of difference in sequences and series with examples.<\/p>\n

Let’s begin –<\/p>\n

Method of Difference<\/h2>\n

Some times the \\(n^{th}\\) term of a sequence or a series can not be determined by the method, we have discussed earlier. So we compute the difference between the successive terms of given sequence for obtained the \\(n^{th}\\) terms.<\/p>\n

If \\(T_1\\), \\(T_2\\), \\(T_3\\),…….,\\(T_n\\) are the terms of a sequence then some times the terms \\(T_2\\) – \\(T_1\\),
\\(T_3\\) – \\(T_1\\)…… constitute an AP\/GP. \\(n^{th}\\) term of the series is determined & the sum to n terms of the sequence can easily be obtained.<\/p>\n

Case 1 :<\/strong><\/p>\n

(a) If difference series are in A.P., then<\/p>\n

Let \\(T_n\\) = \\(an^2\\) + bn + c, where a, b, c are constant<\/p>\n

(b) If difference of difference series are in A.P.<\/p>\n

Let \\(T_n\\) = \\(an^3\\) + \\(bn^2\\) + cn + d, where a, b, c, d are constant<\/p>\n

Case 2 :<\/strong><\/p>\n

(a) If difference are in G.P., then<\/p>\n

Let \\(T_n\\) = \\(ar^n\\) + b, where r is common ratio & a, b are constant<\/p>\n

(b) If difference of difference are in G.P., then<\/p>\n

Let \\(T_n\\) = \\(ar^n\\) + bn + c, where r is common ratio & a, b, c are constant<\/p>\n

Determine constant by putting n = 1, 2, 3 ……. n and putting the value of \\(T_1\\), \\(T_2\\), \\(T_3\\)……. and sum of series \\(S_n\\) = \\({\\sum}T_n\\)<\/p>\n\n\n

Example : <\/span>Find the sum to n terms of the series : 3 + 15 + 35 + 63 + …..<\/p>\n

Solution : <\/span>The difference between the successive terms are 15 – 3 = 12, 35 – 15 = 20, 63 – 35 = 28, …. Clearly,these differences are in AP.

\nLet \\(T_n\\) be the nth term and \\(S_n\\) denote the sum to n terms of the given series

\nThen, \\(S_n\\) = 3 + 15 + 35 + 63 + ….. + \\(T_{n-1}\\) + \\(T_n\\) ……(i)

\nAlso, \\(S_n\\) =     3 + 15 + 35 + ….. + \\(T_{n-1}\\) + \\(T_n\\) ……(ii)

\nSubtracting (ii) from (i), we get

\n0 = 3 + [12 + 20 + 28 + …… + \\(T_{n-1}\\) + \\(T_n\\)] – \\(T_n\\)

\n\\(\\implies\\) \\(T_n\\) = 3 + \\((n-1)\\over 2\\){2*12+(n-1-1)*8} = 3 + (n-1){12+4n-8}

\n\\(\\implies\\) \\(T_n\\) = 3 + (n-1)(4n+4) = \\(4n^2 – 1\\)

\n\\(\\therefore\\) \\(S_n\\) = \\(\\sum_{k=1}^{n}\\) \\(T_k\\) = \\(\\sum_{k=1}^{n}\\) (\\(4k^2 – 1\\)) = 4\\(\\sum_{k=1}^{n}\\)\\(k^2\\) – \\(\\sum_{k=1}^{n}\\) 1

\n\\(\\implies\\) \\(S_n\\) = 4{\\(n(n+1)(2n+1)\\over 6\\)} – n = \\(n\\over 3\\) (\\(4n^2 + 6n – 1\\))

<\/p>\n\n\n

\n

Related Questions<\/h3>\n

Find the sum of n terms of the series 3 + 7 + 14 + 24 + 37 + \u2026\u2026.<\/a><\/p>\n

Find the sum of n terms of the series 1.3.5 + 3.5.7 + 5.7.9 + \u2026\u2026<\/a><\/p>\n\n\n

\n
Previous – Sum of GP to Infinity \u2013 Example & Proof<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn method of difference in sequences and series with examples. Let’s begin – Method of Difference Some times the \\(n^{th}\\) term of a sequence or a series can not be determined by the method, we have discussed earlier. So we compute the difference between the successive terms of given sequence for obtained …<\/p>\n

Method of Difference – Sequences and 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":[125,129,127,126],"yoast_head":"\nMethod of Difference - Sequences and Series - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn method of difference in sequences and series 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\/method-of-difference-sequences-series\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Method of Difference - 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