## 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.