For two numbers [math]x[/math] and [math]y[/math], let [math]\,x, \, a, \, y\,[/math] be a sequence of three numbers. If [math]\,x, \, a, \, y\,[/math] is an arithmetic progression then [math]a[/math] is called arithmetic mean. If [math]\,x, \, a, \, y\,[/math] is a geometric progression then [math]a[/math] is called geometric mean. If [math]\,x, \, a, \, y\,[/math] form a harmonic progression then [math]a[/math] is called harmonic mean.

Let [math]AM[/math] = arithmetic mean, [math]GM[/math] = geometric mean, and [math]HM[/math] = harmonic mean. The relationship between the three is given by the formula

Below is the derivation of this relationship.