{"id":4022,"date":"2021-08-13T21:04:55","date_gmt":"2021-08-13T21:04:55","guid":{"rendered":"https:\/\/mathemerize.com\/?p=4022"},"modified":"2021-11-17T17:38:30","modified_gmt":"2021-11-17T12:08:30","slug":"harmonic-progression-formulas","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/harmonic-progression-formulas\/","title":{"rendered":"Sum of Harmonic Progression | HP Series"},"content":{"rendered":"

Here you will learn what is harmonic progression (hp) and sum of harmonic progression (hp series).<\/p>\n

Let’s begin –<\/p>\n

Sum of Harmonic Progression<\/h2>\n

A sequence is said to be in H.P. if the reciprocal of its terms are in A.P. If the sequence \\(a_1\\) + \\(a_2\\) + \\(a_3\\) + ……… + \\(a_n\\) is an HP then \\(1\\over a_1\\), \\(1\\over a_2\\)……… \\(1\\over a_n\\),is an AP. Here we do not have the formula for the sum of the n terms of an HP.<\/p>\n

The general form of a harmonic progression is \\(1\\over a\\), \\(1\\over {a+d}\\), \\(1\\over {a+2d}\\), \\(1\\over {a+(n-1)d}\\).<\/p>\n

Note :<\/strong><\/p>\n

No term of any H.P can be zero.<\/p>\n

If a, b, c are in H.P, then b = \\(2ac\\over{a+c}\\)<\/p>\n\n\n

Example : <\/span> The sum of three numbers are in H.P.. is 37 and the sum of their reciprocals is 1\/4. Find the numbers.<\/p>\n

Solution : <\/span>Three numbers in H.P. can be taken as

\n\t\t       \\(1\\over {a-d}\\), \\(1\\over a\\), \\(1\\over {a+d}\\)

\n\t\t then   \\(1\\over {a-d}\\)+\\(1\\over a\\)+\\(1\\over {a+d}\\) = 37    …….(i)

\n\t\t and    a – d + a + a + d = \\(1\\over 4\\)    \\(\\implies\\)    a = \\(1\\over 12\\)

\n\t\t from(i),    \\(12\\over {1-12d}\\)+12+\\(12\\over {1+12d}\\) = 37    \\(\\implies\\)    \n\t\t \\(12\\over {1-12d}\\)+\\(12\\over {1+12d}\\) = 25

\n\t\t =>    \\(24\\over {1-144d^2}\\) = 25    \\(\\implies\\)    1-\\(144d^2\\) = \\(24\\over 25\\)    \\(\\implies\\)   \n\t\t \\(d^2\\) = \\(1\\over{25\\times144}\\)

\n\t\t \\(\\therefore\\)    d = \\(\\pm\\)\\(1\\over 60\\)

\n\t\t \\(\\therefore\\)    a-d, a, a+d are \\(1\\over 15\\), \\(1\\over 12\\), \\(1\\over 10\\) or \\(1\\over 10\\), \\(1\\over 12\\), \\(1\\over 15\\)\n\t\t

Hence, three numbers in H.P are 15,12,10 or 10,12,15.

<\/p>\n\n\n

Hope you learnt what is harmonic progression (hp) and sum of harmonic progression. To learn more practice more questions and get ahead in competition. Good Luck!<\/p>\n\n\n

\n
Next – Relation Between Arithmetic Geometric and Harmonic mean<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Sum of GP Series Formula | Properties of GP<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn what is harmonic progression (hp) and sum of harmonic progression (hp series). Let’s begin – Sum of Harmonic Progression A sequence is said to be in H.P. if the reciprocal of its terms are in A.P. If the sequence \\(a_1\\) + \\(a_2\\) + \\(a_3\\) + ……… + \\(a_n\\) is an HP …<\/p>\n

Sum of Harmonic Progression | HP Series<\/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":[167,168,169,166,170],"yoast_head":"\nSum of Harmonic Progression | HP Series - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn what is harmonic progression (hp) and sum of harmonic progression series and harmonic progression formulas.\" \/>\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\/harmonic-progression-formulas\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Sum of Harmonic Progression | HP Series - 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