{"id":6043,"date":"2021-10-04T23:36:59","date_gmt":"2021-10-04T18:06:59","guid":{"rendered":"https:\/\/mathemerize.com\/?p=6043"},"modified":"2021-10-04T23:37:30","modified_gmt":"2021-10-04T18:07:30","slug":"sequences-series-examples","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/sequences-series-examples\/","title":{"rendered":"Sequences and Series Examples"},"content":{"rendered":"

Here you will learn some sequences and series examples for better understanding of sequences and series concepts<\/a><\/span>.<\/p>\n\n\n

\n
\n\n \n

Example 1 : <\/span> If \\({\\sum}_{r=1}^{n\u200e} T_r\\) = \\(n\\over 8\\) (n + 1)(n + 2)(n + 3), then find \\({\\sum}_{r=1}^{n\u200e} \\)\\(1\\over T_r\\)<\/p>\n

Solution : <\/span>    \\(\\because\\)     \\(T_n\\) = \\(S_n – S_{n-1}\\)

\n\t\t\t = \\({\\sum}_{r=1}^{n\u200e} T_r\\) – \\({\\sum}_{r=1}^{n\u200e – 1} T_r\\)

\n\t\t\t = \\(n(n+1)(n+2)(n+3)\\over 8\\) – \\((n-1)(n)(n+1)(n+2)\\over 8\\)

\n\t\t\t = \\(n(n+1)(n+2)\\over 8\\)[(n+3) – (n-1)] = \\(n(n+1)(n+2)\\over 8\\)(4)

\n\t\t\t \\(T_n\\) = \\(n(n+1)(n+2)\\over 2\\)

\n\t\t\t \\(\\implies\\) \\(1\\over T_n\\) = \\(2\\over n(n+1)(n+2)\\) = \\((n+2)-n\\over n(n+1)(n+2)\\)

\n\t\t\t = \\(1\\over n(n+1)\\) – \\(1\\over (n+1)(n+2)\\)

\n\t\t\t Let \\(V_n\\) = \\(1\\over n(n+1)\\)

\n\t\t\t \\(\\therefore\\) \\(1\\over T_n\\) = \\(V_n\\) – \\(V_{n+1}\\)

\n\t\t\t Putting n = 1, 2, 3, …… n

\n\t\t\t \\(\\implies\\) \\(1\\over T_1\\) + \\(1\\over T_2\\) + \\(1\\over T_3\\) + ….. + \\(1\\over T_n\\) = \\(V_1\\) – \\(V_{n+1}\\)

\n\t\t\t = \\({\\sum}_{r=1}^{n\u200e} \\)\\(1\\over T_r\\) = \\(n^2 + 3n\\over 2(n+1)(n+2)\\)\n\t\t\t<\/p>\n


\n\n

Example 2 : <\/span> Example 2: Find the sum of n terms of the series 1.3.5 + 3.5.7 + 5.7.9 + ……<\/p>\n

Solution : <\/span>The \\(n^{th}\\) term is (2n-1)(2n+1)(2n+3)

\n\t\t \\(T_n\\) = (2n-1)(2n+1)(2n+3)

\n\t\t \\(T_n\\) = \\(1\\over 8\\)(2n-1)(2n+1)(2n+3){(2n+5) – (2n-3)}

\n\t\t = \\(1\\over 8\\)(\\(V_n\\) – \\(V_{n-1}\\))

\n\t\t \\(S_n\\) = \\({\\sum}_{r=1}^{n\u200e} T_n\\) = \\(1\\over 8\\)(\\(V_n\\) – \\(V_0\\))

\n\t\t \\(\\therefore\\)     \\(S_n\\) = \\((2n-1)(2n+1)(2n+3)(2n+5)\\over 8\\) + \\(15\\over 8\\)

\n\t\t = \\(n(2n^3 + 8n^2 + 7n – 2)\\)\n
\n <\/p>

Practice these given sequences and series examples to test your knowledge on concepts of sequences and series.<\/p>\n <\/div>\n <\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn some sequences and series examples for better understanding of sequences and series concepts. Example 1 : If \\({\\sum}_{r=1}^{n\u200e} T_r\\) = \\(n\\over 8\\) (n + 1)(n + 2)(n + 3), then find \\({\\sum}_{r=1}^{n\u200e} \\)\\(1\\over T_r\\) Solution :     \\(\\because\\)     \\(T_n\\) = \\(S_n – S_{n-1}\\) = \\({\\sum}_{r=1}^{n\u200e} T_r\\) – \\({\\sum}_{r=1}^{n\u200e …<\/p>\n

Sequences and Series Examples<\/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":[],"yoast_head":"\nSequences and Series Examples - Mathemerize<\/title>\n<meta name=\"description\" content=\"Solve these Sequences and Series examples to practice and test your knowledge of Sequences and Series concepts.\" \/>\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\/sequences-series-examples\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Sequences and Series Examples - 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