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.