When to use the geometric mean?

This is better documented elsewhere, but I could not find a quick reference on the web as to when you’d want to use the geometric mean instead of the arithmetic (usual) mean.

• Suppose that I’m 30% richer than last year, but last year I was 20% richer than the year before… what is the average growth? Well, my current wealth is 1.3 * 1.2 * w if w is my wealth two years ago. I can expect that if t is the average growth factor over the last two years, then my current wealth is t * t * w. Setting t = 1.25 is the wrong answer. In such a case, choosing t = sqrt(1.3 * 1.2) solves the problem.
• Another case where the geometric mean makes sense is when you are stuck averaging numbers that are not comparable like the time necessary to build a data cube, versus the average query time. Indeed, if a and b are two numbers and a is much smaller than b, then (2a +b)/2 is about the same as (a+b)/2. One component of your system is significantly worse and yet, you get the same average performance? That’s wrong. Computing sqrt (2ab) seems to make much more sense.