For positive numbers and , the **geometric mean** is .

e.g. If an investment earns 25% in the first year (i.e. the amount is multiplied by ) and 80% in the second year () then the average annual rate of return is , since .

For positive ,

The two sides are equal when

For :

as shown in the following diagrams:

**1.**

i.e. when , the rectangle is smaller than the area of the two triangular half-squares

**2.**

Why is the height ?

By similar triangles,

## Harmonic mean

If we travel a distance at rate in time and make the return trip at rate in time , then . What’s the average rate for the full trip?

and therefore:

This average rate is the harmonic mean of the two rates, and less than the average – the arithmetic mean – when the rates differ, because more time is spent at the lower speed.

## HM GM AM

i.e. .

### Some other means

Root mean square (RMS) :~~+++~~

Contraharmonic mean :~~+++~~

Heronian mean :~~+++~~

Logarithmic mean :~~+++~~

Identric mean :~~+++~~

**References**

Alsina, Nelsen – When Less Is More

Nelsen – Proofs Without Words, I & II