the arithmetic, geometric and harmonic mean

For positive numbers $a$ and $b$ , the geometric mean is $\sqrt{ab}$ .
e.g. If an investment earns 25% in the first year (i.e. the amount is multiplied by $a=1.25$ ) and 80% in the second year ( $b=1.8$ ) then the average annual rate of return $r$ is $\sqrt{ab}=1.5$ , since $ab=r^2$ .

For positive $a,b,c\dots$ , $\displaystyle\sqrt[n]{abc\dots}\les\frac{a+b+c+\dots}{n}$
The two sides are equal when $a=b=c=\dots$
For $n=2$ :

$\sqrt{ab}\les\frac{a+b}{2}\quad\text{(the AM-GM inequality)}$

as shown in the following diagrams:

1.

i.e. when $a\neq b$ , the rectangle $\sqrt{a}\sqrt{b}$ is smaller than the area of the two triangular half-squares $\displaystyle\frac{a}{2}+\frac{b}{2}$

2.

Why is the height $h=\sqrt{ab}$ ?
By similar triangles, $\quad$ $\displaystyle \frac{h}{a}=\frac{b}{h} \quad \to \quad h^2=ab \quad \to \quad h=\sqrt{ab}$

Harmonic mean

If we travel a distance $d$ at rate $r_1$ in time $t_1$ and make the return trip at rate $r_2$ in time $t_2$ , then $r_1t_1=d=r_2t_2$ . What’s the average rate for the full trip?
$2d=r(t_1+t_2)$ and therefore:

$r=\frac{2d}{t_1+t_2}=\frac{2d}{\frac{d}{r_1}+\frac{d}{r_2}}=\frac{2r_1r_2}{r_1+r_2}$

This average rate is the harmonic mean of the two rates, and less than the average – the arithmetic mean $\displaystyle\frac{r_1+r_2}{2}$ – when the rates differ, because more time is spent at the lower speed.