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 = GM²

x, \, AM, \, y \, \to \, arithmetic progression

Taking the common difference of arithmetic progression,
AM - x = y - AM
y = 2 \, AM - x \, \to \, Equation (1)

x, \, GM, \, y \, \to \, geometric progression

The common ratio of this geometric progression is
\dfrac{GM}{x} = \dfrac{y}{GM}
y = \dfrac{GM^2}{x} \, \to \, Equation (2)

Equate Equations (1) and (2)
2 \, AM - x = \dfrac{GM^2}{x}
2x \, AM - x^2 = GM^2
x^2 - 2x \, AM = -GM^2 \, \to \, Equation (3)

x, \, HM, \, y \, \to \, harmonic progression
\dfrac{1}{x}, \, \dfrac{1}{HM}, \, \dfrac{1}{y} \, \to \, 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}{x} = \dfrac{1}{y}
\dfrac{2x - HM}{x \, HM} = \dfrac{1}{y}
y = \dfrac{x \, HM}{2x - HM} \, \to \, Equation (4)

Equate Equations (1) and (4)
2 \, AM - x = \dfrac{x \, HM}{2x - HM}
(2 \, AM - x)(2x - HM) = x \, HM
4x \, AM - 2 \, AM \, HM - 2x^2 + x \, HM = x \, HM
4x \, AM - 2 \, AM \, HM - 2x^2 = 0
2x \, AM - AM \, HM - x^2 = 0
x^2 - 2x \, AM = -AM \, HM \, \to \, Equation (5)

Equate Equations (3) and (5)
-AM \, HM = -GM^2