{"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":"

Solution :<\/h2>\n

Let a, ar, \\(ar^2\\) are in GP (r &gt; 1)<\/p>\n

According to the question, a, 2ar, \\(ar^2\\) in AP.<\/p>\n

\\(\\implies\\)\u00a0 4ar = a + \\(ar^2\\)<\/p>\n

\\(\\implies\\) \\(r^2\\) – 4r + 1 = 0<\/p>\n

\\(\\implies\\) r = \\(2 \\pm \\sqrt{3}\\)<\/p>\n

Hence, r = \\(2 + \\sqrt{3}\\)\u00a0 \u00a0 [ \\(\\because\\)\u00a0 AP is increasing]<\/p>\n

\n

Similar Questions<\/h3>\n

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

The sum of first 20 terms of the sequence 0.7, 0.77, 0.777, \u2026\u2026. , is<\/a><\/p>\n

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

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

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":"

Solution : Let a, ar, \\(ar^2\\) are in GP (r &gt; 1) According to the question, a, 2ar, \\(ar^2\\) in AP. \\(\\implies\\)\u00a0 4ar = a + \\(ar^2\\) \\(\\implies\\) \\(r^2\\) – 4r + 1 = 0 \\(\\implies\\) r = \\(2 \\pm \\sqrt{3}\\) Hence, r = \\(2 + \\sqrt{3}\\)\u00a0 \u00a0 [ \\(\\because\\)\u00a0 AP is increasing] Similar Questions …<\/p>\n

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 »<\/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":"\nThree 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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