{"id":6872,"date":"2021-10-21T21:30:05","date_gmt":"2021-10-21T16:00:05","guid":{"rendered":"https:\/\/mathemerize.com\/?p=6872"},"modified":"2021-11-17T15:34:14","modified_gmt":"2021-11-17T10:04:14","slug":"find-the-sum-of-n-terms-of-the-series-1-3-5-3-5-7-5-7-9","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/find-the-sum-of-n-terms-of-the-series-1-3-5-3-5-7-5-7-9\/","title":{"rendered":"Find the sum of n terms of the series 1.3.5 + 3.5.7 + 5.7.9 + &#8230;&#8230;"},"content":{"rendered":"<h2>Solution :<\/h2>\n<p>By using <a href=\"https:\/\/mathemerize.com\/method-of-difference-sequences-series\/\">method of differences<\/a>,<\/p>\n<p>The \\(n^{th}\\) term is (2n-1)(2n+1)(2n+3)<\/p>\n<p>\\(T_n\\) = (2n-1)(2n+1)(2n+3)<\/p>\n<p>\\(T_n\\) = \\(1\\over 8\\)(2n-1)(2n+1)(2n+3){(2n+5) &#8211; (2n-3)}<\/p>\n<p>= \\(1\\over 8\\)(\\(V_n\\) &#8211; \\(V_{n-1}\\))<\/p>\n<p>\\(S_n\\) = \\({\\sum}_{r=1}^{n\u200e} T_n\\) = \\(1\\over 8\\)(\\(V_n\\) &#8211; \\(V_0\\))<\/p>\n<p>\\(\\therefore\\)\u00a0 \\(S_n\\) = \\((2n-1)(2n+1)(2n+3)(2n+5)\\over 8\\) + \\(15\\over 8\\)<\/p>\n<p>= \\(n(2n^3 + 8n^2 + 7n &#8211; 2)\\)<\/p>\n<hr \/>\n<h3 style=\"text-align: center;\">Similar Questions<\/h3>\n<p><a class=\"row-title\" href=\"https:\/\/mathemerize.com\/find-the-sum-of-n-terms-of-the-series-3-7-14-24-37\/\" aria-label=\"\u201cFind the sum of n terms of the series 3 + 7 + 14 + 24 + 37 + \u2026\u2026.\u201d (Edit)\">Find the sum of n terms of the series 3 + 7 + 14 + 24 + 37 + \u2026\u2026.<\/a><\/p>\n<p><a class=\"row-title\" href=\"https:\/\/mathemerize.com\/find-the-sum-to-n-terms-of-the-series-3-15-35-63\/\" aria-label=\"\u201cFind the sum to n terms of the series : 3 + 15 + 35 + 63 + \u2026..\u201d (Edit)\">Find the sum to n terms of the series : 3 + 15 + 35 + 63 + \u2026..<\/a><\/p>\n<p><a class=\"row-title\" href=\"https:\/\/mathemerize.com\/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\/\" aria-label=\"\u201cIf 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\u201d (Edit)\">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<p><a class=\"row-title\" href=\"https:\/\/mathemerize.com\/if-x-y-and-z-are-in-ap-and-tan-1x-tan-1y-and-tan-1z-are-also-in-ap-then\/\" aria-label=\"\u201cIf x, y and z are in AP and \\(tan^{-1}x\\), \\(tan^{-1}y\\) and \\(tan^{-1}z\\) are also in AP, then\u201d (Edit)\">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<p><a class=\"row-title\" href=\"https:\/\/mathemerize.com\/three-positive-integers-form-an-increasing-gp-if-the-middle-term-in-this-gp-is-doubled-then-new-numbers-are-in-ap-then-the-common-ratio-of-gp-is\/\" aria-label=\"\u201cThree positive integers form an increasing GP. If the middle term in this GP is doubled, then new numbers are in AP. Then the common ratio of GP is\u201d (Edit)\">Three positive integers form an increasing GP. If the middle term in this GP is doubled, then new numbers are in AP. Then the common ratio of GP is<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Solution : By using method of differences, The \\(n^{th}\\) term is (2n-1)(2n+1)(2n+3) \\(T_n\\) = (2n-1)(2n+1)(2n+3) \\(T_n\\) = \\(1\\over 8\\)(2n-1)(2n+1)(2n+3){(2n+5) &#8211; (2n-3)} = \\(1\\over 8\\)(\\(V_n\\) &#8211; \\(V_{n-1}\\)) \\(S_n\\) = \\({\\sum}_{r=1}^{n\u200e} T_n\\) = \\(1\\over 8\\)(\\(V_n\\) &#8211; \\(V_0\\)) \\(\\therefore\\)\u00a0 \\(S_n\\) = \\((2n-1)(2n+1)(2n+3)(2n+5)\\over 8\\) + \\(15\\over 8\\) = \\(n(2n^3 + 8n^2 + 7n &#8211; 2)\\) Similar Questions Find the &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathemerize.com\/find-the-sum-of-n-terms-of-the-series-1-3-5-3-5-7-5-7-9\/\"> <span class=\"screen-reader-text\">Find the sum of n terms of the series 1.3.5 + 3.5.7 + 5.7.9 + &#8230;&#8230;<\/span> Read More &raquo;<\/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":[43,57],"tags":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v20.12 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Find the sum of n terms of the series 1.3.5 + 3.5.7 + 5.7.9 + ......<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, 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5.7.9 + &#8230;&#8230; Read More &raquo;","og_url":"https:\/\/mathemerize.com\/find-the-sum-of-n-terms-of-the-series-1-3-5-3-5-7-5-7-9\/","og_site_name":"Mathemerize","article_published_time":"2021-10-21T16:00:05+00:00","article_modified_time":"2021-11-17T10:04:14+00:00","author":"mathemerize","twitter_card":"summary_large_image","twitter_misc":{"Written by":"mathemerize","Est. reading time":"1 minute"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/mathemerize.com\/find-the-sum-of-n-terms-of-the-series-1-3-5-3-5-7-5-7-9\/#article","isPartOf":{"@id":"https:\/\/mathemerize.com\/find-the-sum-of-n-terms-of-the-series-1-3-5-3-5-7-5-7-9\/"},"author":{"name":"mathemerize","@id":"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df"},"headline":"Find the sum of n terms of the series 1.3.5 + 3.5.7 + 5.7.9 + 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