# Relationship Between Arithmetic Mean, Harmonic Mean, and Geometric Mean of Two Numbers

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

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 = 2AM from Equation (1) and xy = GM^{2} from Equation (2) to Equation (3)

$GM^2 = AM \times HM$ → *Okay!*