For two numbers *x* and *y*, let *x*, *a*, *y* be a sequence of three numbers. If *x*, *a*, *y* is an arithmetic progression then '*a*' is called ** arithmetic mean**. If

*x*,

*a*,

*y*is a geometric progression then '

*a*' is called

**. If**

*geometric mean**x*,

*a*,

*y*form a harmonic progression then '

*a*' is called

**.**

*harmonic mean*Let *AM* = arithmetic mean, *GM* = geometric mean, and *HM* = harmonic mean. The relationship between the three is given by the formula

Below is the derivation of this relationship.

**Derivation of AM × HM = GM^{2}**

Arithmetic Progression

Taking the common difference of arithmetic progression,

$AM - x = y - AM$

$x + y = 2 \, AM$ ← Equation (1)

Geometric Progression

The common ratio of this geometric progression is

$\dfrac{GM}{x} = \dfrac{y}{GM}$

$xy = GM^2$ ← Equation (2)

Harmonic Progression

$\dfrac{1}{x}, \, \dfrac{1}{HM}, \, \dfrac{1}{y}$ ← the reciprocal of each term will form an arithmetic progression

The common difference is

$\dfrac{1}{HM} - \dfrac{1}{x} = \dfrac{1}{y} - \dfrac{1}{HM}$

$\dfrac{2}{HM} = \dfrac{1}{y} + \dfrac{1}{x}$

$\dfrac{2}{HM} = \dfrac{x + y}{xy}$ ← Equation (3)

Substitute *x* + *y* = 2*AM* from Equation (1) and *xy* = *GM*^{2} from Equation (2) to Equation (3)

$GM^2 = AM \times HM$ ← *Okay!*