# 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 $$a_1$$ = $$a_2$$ =…..= $$a_n$$.