# the arithmetic, geometric and harmonic mean

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,

3.

4.

5.

## 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.

## HMGMAM

i.e. .

The video shows points in .

### 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