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
as shown in the following diagrams:
i.e. when , the rectangle is smaller than the area of the two triangular half-squares
Why is the height ?
By similar triangles,
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?
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
Some other means
Root mean square (RMS) :
Contraharmonic mean :
Heronian mean :
Logarithmic mean :
Identric mean :
Alsina, Nelsen – When Less Is More
Nelsen – Proofs Without Words, I & II