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 geometric mean. If 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

$AM \times HM = GM^2$

Below is the derivation of this relationship.

Derivation of AM × HM = GM2
Arithmetic Progression

$x, \, AM, \, y$   ←   arithmetic progression

Taking the common difference of arithmetic progression,
$AM - x = y - AM$

$x + y = 2 \, AM$   ←   Equation (1)

Geometric Progression

$x, \, GM, \, y$   ←   geometric progression

The common ratio of this geometric progression is
$\dfrac{GM}{x} = \dfrac{y}{GM}$

$xy = GM^2$   ←   Equation (2)

Harmonic Progression

$x, \, HM, \, y$   ←   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 = 2AM from Equation (1) and xy = GM2 from Equation (2) to Equation (3)

$\dfrac{2}{HM} = \dfrac{2 \, AM}{GM^2}$

$GM^2 = AM \times HM$   ←   Okay!

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