Relation Between Arithmetic Geometric and Harmonic mean

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 a & c.

So A.M. of a and c = \({a+b}\over 2\) = b

n-Arithmetic Means between two numbers :

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 = \({b-a}\over {n+1}\)

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

Note :

Sum of n A.M’s inserted between a & b is equal to n times the single A.M. between a & b.

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

Geometric Mean – Formula for Geometric Mean

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

So G.M. of a and c = \(\sqrt{ac}\) = b

n-Geometric Means between two numbers :

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 = \((b/a)^{1\over {n+1}}\)

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

Note :

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

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

Harmonic Mean – Formula for harmonic mean

If a, b, c are in H.P., then b is H.M. between a & c.

So H.M. of a and c = \(2ac\over{a+c}\) = b

Insertion of ‘n’ HM’s between a and b :

a, \(H_1\), \(H_2\), \(H_3\),……,\(H_n\), b \(\rightarrow\) H.P

\(1\over a\), \(1\over{H_1}\), \(1\over{H_2}\), \(1\over{H_3}\),……..,\(1\over{H_n}\), \(1\over b\) \(\rightarrow\) A.P.

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

\(1\over{H_n}\) = \(1\over a\) + n(\({{1\over a}-{1\over b}}\over {n+1}\))

Relation Between Arithmetic Geometric and Harmonic mean

(i)  If A, G, H, are respectively A.M., G.M., H.M. between two positive number a & b then

(a)  \(G^2\) = AH (A, G, H constitute a GP)

(b)  \(A \ge G \ge H\)

(c)  A = G = H \(\Leftrightarrow\) a = b

(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 = \({a_1 + a_2 + a_3 + ……… + a_n}\over n\)

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

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

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