Question

If $a,H,b$ are in Harmonic Progression and $H$ is called the harmonic mean between $a$ and $b$, then prove that, $H=\frac{2ab}{a+b}$.

Answer 1

Given that:

$a,H,b$ are in H.P.

To prove:

$H=\frac{2ab}{a+b}$

Solution:

If $a,H,b$ are in H.P.

then, $\frac{1}{a},\frac{1}{H},\frac{1}{b}$ are in A.P.

So,

$\frac{1}{H}-\frac{1}{a}=\frac{1}{b}-\frac{1}{H}$

or, $\frac{2}{H}=\frac{1}{a}+\frac{1}{b}$

or, $\frac{2}{H}=\frac{a+b}{ab}$

or, $\frac{1}{H}=\frac{a+b}{2ab}$

or, $H=\frac{2ab}{a+b}$

Hence, proved.