{"id":7038,"date":"2021-10-22T15:47:16","date_gmt":"2021-10-22T10:17:16","guid":{"rendered":"https:\/\/mathemerize.com\/?p=7038"},"modified":"2021-10-25T02:22:36","modified_gmt":"2021-10-24T20:52:36","slug":"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","status":"publish","type":"post","link":"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\/","title":{"rendered":"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"},"content":{"rendered":"<h2>Solution :<\/h2>\n<p>Let a, ar, \\(ar^2\\) are in GP (r &amp;gt; 1)<\/p>\n<p>According to the question, a, 2ar, \\(ar^2\\) in AP.<\/p>\n<p>\\(\\implies\\)\u00a0 4ar = a + \\(ar^2\\)<\/p>\n<p>\\(\\implies\\) \\(r^2\\) &#8211; 4r + 1 = 0<\/p>\n<p>\\(\\implies\\) r = \\(2 \\pm \\sqrt{3}\\)<\/p>\n<p>Hence, r = \\(2 + \\sqrt{3}\\)\u00a0 \u00a0 [ \\(\\because\\)\u00a0 AP is increasing]<\/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\/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<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\/find-the-sum-of-n-terms-of-the-series-1-3-5-3-5-7-5-7-9\/\" aria-label=\"\u201cFind the sum of n terms of the series 1.3.5 + 3.5.7 + 5.7.9 + \u2026\u2026\u201d (Edit)\">Find the sum of n terms of the series 1.3.5 + 3.5.7 + 5.7.9 + \u2026\u2026<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Solution : Let a, ar, \\(ar^2\\) are in GP (r &amp;gt; 1) According to the question, a, 2ar, \\(ar^2\\) in AP. \\(\\implies\\)\u00a0 4ar = a + \\(ar^2\\) \\(\\implies\\) \\(r^2\\) &#8211; 4r + 1 = 0 \\(\\implies\\) r = \\(2 \\pm \\sqrt{3}\\) Hence, r = \\(2 + \\sqrt{3}\\)\u00a0 \u00a0 [ \\(\\because\\)\u00a0 AP is increasing] Similar Questions &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" 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\/\"> <span class=\"screen-reader-text\">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<\/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>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<\/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\/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\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"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\" \/>\n<meta property=\"og:description\" content=\"Solution : Let a, ar, (ar^2) are in GP (r &amp;gt; 1) According to the question, a, 2ar, (ar^2) in AP. 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If the middle term in this GP is doubled, then new numbers are in AP. Then the common ratio of GP is","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"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\/","og_locale":"en_US","og_type":"article","og_title":"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","og_description":"Solution : Let a, ar, (ar^2) are in GP (r &amp;gt; 1) According to the question, a, 2ar, (ar^2) in AP. (implies)\u00a0 4ar = a + (ar^2) (implies) (r^2) &#8211; 4r + 1 = 0 (implies) r = (2 pm sqrt{3}) Hence, r = (2 + sqrt{3})\u00a0 \u00a0 [ (because)\u00a0 AP is increasing] Similar Questions &hellip; Three positive integers form an increasing GP. If the middle term in this GP is doubled, then new numbers are in AP. 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