{"id":4074,"date":"2021-08-15T21:51:54","date_gmt":"2021-08-15T21:51:54","guid":{"rendered":"https:\/\/mathemerize.com\/?p=4074"},"modified":"2021-11-17T01:20:34","modified_gmt":"2021-11-16T19:50:34","slug":"arithmetic-geometric-and-harmonic-mean","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/arithmetic-geometric-and-harmonic-mean\/","title":{"rendered":"Relation Between Arithmetic Geometric and Harmonic mean"},"content":{"rendered":"

Here you will learn formula for arithmetic geometric and harmonic mean and relation between arithmetic geometric and harmonic mean.<\/p>\n

Let’s begin –<\/p>\n

Arithmetic Mean Formula<\/h2>\n

If three terms are in A.P. then the middle term is called the A.M. between the other two, so if a, b, c are in A.P., b is A.M. of a & c.<\/p>\n

\n

So A.M. of a and c = \\({a+b}\\over 2\\) = b<\/p>\n<\/blockquote>\n

n-Arithmetic Means between two numbers<\/strong> :<\/h4>\n

If a, b be any two given numbers & a, \\(A_1\\), \\(A_2\\)……\\(A_n\\), b are in AP, then \\(A_1\\), \\(A_2\\)……\\(A_n\\) are the ‘n’ A.M’s between a & b then. \\(A_1\\) = a + d, \\(A_2\\) = a + 2d ,……., \\(A_n\\) = a + nd or b – d, where d = \\({b-a}\\over {n+1}\\)<\/p>\n

\\(\\implies\\) \\(A_1\\) = a + \\({b-a}\\over {n+1}\\), \\(A_2\\)= a + \\(2({b-a})\\over {n+1}\\),…..<\/p>\n

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

Sum of n A.M’s inserted between a & b is equal to n times the single A.M. between a & b.<\/p>\n

i.e. \\({\\sum_{r=1}^{n}A_r}\\) = nA where A is the single A.M. between a & b.<\/p>\n

Geometric Mean – Formula for Geometric Mean<\/h2>\n

If a, b, c are in G.P., then b is the G.M. between a & c, \\(b^2\\) = ac.<\/p>\n

\n

So G.M. of a and c = \\(\\sqrt{ac}\\) = b<\/p>\n<\/blockquote>\n

n-Geometric Means between two numbers<\/strong> :<\/h4>\n

If a, b be any two given positive numbers & a, \\(G_1\\), \\(G_2\\)…… \\(G_n\\), b are in G.P. Then \\(G_1\\), \\(G_2\\)……\\(G_n\\) are the ‘n’ G.M’s between a & b, where b = \\(ar^{n+1}\\) => r = \\((b\/a)^{1\\over {n+1}}\\)<\/p>\n

\\(G_1\\) = a\\((b\/a)^{1\\over {n+1}}\\),    \\(G_2\\)= a\\((b\/a)^{2\\over {n+1}}\\),…….   \\(G_n\\)= a\\((b\/a)^{n\\over {n+1}}\\)<\/p>\n

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

The product of n G.Ms between a & b is equal to \\(n^{th}\\) power of the single G.M. between a & b<\/p>\n

i.e. \\(\\prod_{r=1}^{\\infty} G_{r}\\) = \\((G)^n\\) where G is the single G.M. between a & b<\/p>\n

Harmonic Mean – Formula for harmonic mean<\/h2>\n

If a, b, c are in H.P., then b is H.M. between a & c.<\/p>\n

\n

So H.M. of a and c = \\(2ac\\over{a+c}\\) = b<\/p>\n<\/blockquote>\n

Insertion of ‘n’ HM’s between a and b<\/strong> :<\/h4>\n

a, \\(H_1\\), \\(H_2\\), \\(H_3\\),……,\\(H_n\\), b \\(\\rightarrow\\) H.P<\/p>\n

\\(1\\over a\\), \\(1\\over{H_1}\\), \\(1\\over{H_2}\\), \\(1\\over{H_3}\\),……..,\\(1\\over{H_n}\\), \\(1\\over b\\) \\(\\rightarrow\\) A.P.<\/p>\n

\\(1\\over b\\) = \\(1\\over a\\) + (n + 1)D   =>   D = \\({{1\\over a}-{1\\over b}}\\over {n+1}\\)<\/p>\n

\\(1\\over{H_n}\\) = \\(1\\over a\\) + n(\\({{1\\over a}-{1\\over b}}\\over {n+1}\\))<\/p>\n

Relation Between Arithmetic Geometric and Harmonic mean<\/h2>\n

(i)  If A, G, H, are respectively A.M., G.M., H.M. between two positive number a & b then<\/p>\n

\n

(a)  \\(G^2\\) = AH (A, G, H constitute a GP)<\/p>\n

(b)  \\(A \\ge G \\ge H\\)<\/p>\n

(c)  A = G = H \\(\\Leftrightarrow\\) a = b<\/p>\n<\/blockquote>\n

(ii)  Let \\(a_1\\) + \\(a_2\\) + \\(a_3\\) + ……… + \\(a_n\\) be n positive real numbers, then we define their arithmetic mean(A), geometric mean(G) and harmonic mean(H) as A = \\({a_1 + a_2 + a_3 + ……… + a_n}\\over n\\)<\/p>\n

G = \\((a_1 + a_2 + a_3 + ……… + a_n)^{1\\over n}\\) and H = \\(n\\over {1\\over {a_1}} + {1\\over {a_2}} +…..{1\\over {a_n}} \\)<\/p>\n

It can be shown that \\(A \\ge G \\ge H\\). Moreover equality holds at either place if and only if \\(a_1\\) = \\(a_2\\) =…..= \\(a_n\\).<\/p>\n\n\n

\n
Next – What is AGP \u2013 How to Solve AGP Series<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Sum of Harmonic Progression | HP Series<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn formula for arithmetic geometric and harmonic mean and relation between arithmetic geometric and harmonic mean. Let’s begin – Arithmetic Mean Formula If three terms are in A.P. then the middle term is called the A.M. between the other two, so if a, b, c are in A.P., b is A.M. of …<\/p>\n

Relation Between Arithmetic Geometric and Harmonic mean<\/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":[160,164,158,165,159,161,157],"yoast_head":"\nRelation Between Arithmetic Geometric and Harmonic mean<\/title>\n<meta name=\"description\" content=\"In this post you will learn formula for arithmetic geometric and harmonic mean and 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