{"id":7056,"date":"2021-10-22T15:59:20","date_gmt":"2021-10-22T10:29:20","guid":{"rendered":"https:\/\/mathemerize.com\/?p=7056"},"modified":"2021-10-25T02:17:46","modified_gmt":"2021-10-24T20:47:46","slug":"prove-that-the-sum-of-first-n-natural-numbers-is-nn1over-2","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/prove-that-the-sum-of-first-n-natural-numbers-is-nn1over-2\/","title":{"rendered":"Prove that the sum of first n natural numbers is \\(n(n+1)\\over 2\\)"},"content":{"rendered":"<h2>Solution :<\/h2>\n<p>Let \\(S_n\\) = 1 + 2 + 3 + &#8230;.. + n = \\(\\sum_{k=1}^{n} k\\)<\/p>\n<p>Clearly, it is an arithmetic series with first term a = 1, common difference d = 1 and last term l = n.<\/p>\n<p>\\(\\therefore\\) \\(S_n\\) = \\(n\\over 2\\) (1 + n) = \\(n(n+1)\\over 2\\)<\/p>\n<hr \/>\n<h3 style=\"text-align: center;\">Similar Questions<\/h3>\n<p><a class=\"row-title\" href=\"https:\/\/mathemerize.com\/let-a_n-be-the-nth-term-of-an-ap-if-sum_r1100-a_2r-alpha-and-sum_r1100-a_2r-1-beta-then-the-common-difference-of-the-ap-is\/\" aria-label=\"\u201cLet \\(a_n\\) be the nth term of an AP. If \\(\\sum_{r=1}^{100}\\) \\(a_{2r}\\) = \\(\\alpha\\) and \\(\\sum_{r=1}^{100}\\) \\(a_{2r-1}\\) = \\(\\beta\\), then the common difference of the AP is\u201d (Edit)\">Let \\(a_n\\) be the nth term of an AP. If \\(\\sum_{r=1}^{100}\\) \\(a_{2r}\\) = \\(\\alpha\\) and \\(\\sum_{r=1}^{100}\\) \\(a_{2r-1}\\) = \\(\\beta\\), then the common difference of the AP is<\/a><\/p>\n<p><a class=\"row-title\" href=\"https:\/\/mathemerize.com\/a-man-saves-rs-200-in-each-of-the-first-three-months-of-his-service-in-each-of-the-subsequent-months-his-saving-increases-by-rs-40-more-than-the-saving-of-immediately-previous-month-his-total-savin\/\" aria-label=\"\u201cA man saves Rs 200 in each of the first three months of his service. In each of the subsequent months, his saving increases by Rs 40 more than the saving of immediately previous month. His total saving from the start of service will be Rs 11040 after\u201d (Edit)\">A man saves Rs 200 in each of the first three months of his service. In each of the subsequent months, his saving increases by Rs 40 more than the saving of immediately previous month. His total saving from the start of service will be Rs 11040 after<\/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\/the-sum-of-first-20-terms-of-the-sequence-0-7-0-77-0-777-is\/\" aria-label=\"\u201cThe sum of first 20 terms of the sequence 0.7, 0.77, 0.777, \u2026\u2026. , is\u201d (Edit)\">The sum of first 20 terms of the sequence 0.7, 0.77, 0.777, \u2026\u2026. , is<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Solution : Let \\(S_n\\) = 1 + 2 + 3 + &#8230;.. + n = \\(\\sum_{k=1}^{n} k\\) Clearly, it is an arithmetic series with first term a = 1, common difference d = 1 and last term l = n. \\(\\therefore\\) \\(S_n\\) = \\(n\\over 2\\) (1 + n) = \\(n(n+1)\\over 2\\) Similar Questions Let \\(a_n\\) &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathemerize.com\/prove-that-the-sum-of-first-n-natural-numbers-is-nn1over-2\/\"> <span class=\"screen-reader-text\">Prove that the sum of first n natural numbers is \\(n(n+1)\\over 2\\)<\/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>Prove that the sum of first n natural numbers is \\(n(n+1)\\over 2\\)<\/title>\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\/prove-that-the-sum-of-first-n-natural-numbers-is-nn1over-2\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Prove that the sum of first n natural numbers is \\(n(n+1)\\over 2\\)\" \/>\n<meta property=\"og:description\" content=\"Solution : Let (S_n) = 1 + 2 + 3 + &#8230;.. + n = (sum_{k=1}^{n} k) Clearly, it is an arithmetic series with first term a = 1, common difference d = 1 and last term l = n. (therefore) (S_n) = (nover 2) (1 + n) = (n(n+1)over 2) Similar Questions Let (a_n) &hellip; Prove that the sum of first n natural numbers is (n(n+1)over 2) Read More &raquo;\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathemerize.com\/prove-that-the-sum-of-first-n-natural-numbers-is-nn1over-2\/\" \/>\n<meta property=\"og:site_name\" content=\"Mathemerize\" \/>\n<meta property=\"article:published_time\" content=\"2021-10-22T10:29:20+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2021-10-24T20:47:46+00:00\" \/>\n<meta name=\"author\" content=\"mathemerize\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"mathemerize\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"1 minute\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathemerize.com\/prove-that-the-sum-of-first-n-natural-numbers-is-nn1over-2\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/prove-that-the-sum-of-first-n-natural-numbers-is-nn1over-2\/\"},\"author\":{\"name\":\"mathemerize\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df\"},\"headline\":\"Prove that the sum of first n natural numbers is \\\\(n(n+1)\\\\over 2\\\\)\",\"datePublished\":\"2021-10-22T10:29:20+00:00\",\"dateModified\":\"2021-10-24T20:47:46+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathemerize.com\/prove-that-the-sum-of-first-n-natural-numbers-is-nn1over-2\/\"},\"wordCount\":185,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathemerize.com\/#organization\"},\"articleSection\":[\"Maths Questions\",\"Series Questions\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathemerize.com\/prove-that-the-sum-of-first-n-natural-numbers-is-nn1over-2\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathemerize.com\/prove-that-the-sum-of-first-n-natural-numbers-is-nn1over-2\/\",\"url\":\"https:\/\/mathemerize.com\/prove-that-the-sum-of-first-n-natural-numbers-is-nn1over-2\/\",\"name\":\"Prove that the sum of first n natural numbers is \\\\(n(n+1)\\\\over 2\\\\)\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/#website\"},\"datePublished\":\"2021-10-22T10:29:20+00:00\",\"dateModified\":\"2021-10-24T20:47:46+00:00\",\"breadcrumb\":{\"@id\":\"https:\/\/mathemerize.com\/prove-that-the-sum-of-first-n-natural-numbers-is-nn1over-2\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/mathemerize.com\/prove-that-the-sum-of-first-n-natural-numbers-is-nn1over-2\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/mathemerize.com\/prove-that-the-sum-of-first-n-natural-numbers-is-nn1over-2\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/mathemerize.com\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Prove that the sum of first n natural numbers is \\\\(n(n+1)\\\\over 2\\\\)\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/mathemerize.com\/#website\",\"url\":\"https:\/\/mathemerize.com\/\",\"name\":\"Mathemerize\",\"description\":\"Maths Tutorials - 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