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,

→ Equation (1)

Geometric Progression

The common ratio of this geometric progression is

→ Equation (2)

Harmonic Progression

→ the reciprocal of each term will form an arithmetic progression

The common difference is

→ Equation (3)

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

→ *Okay!*

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