Question

If $b$ is the harmonic mean between $a$ and $c$, prove that
$\frac{1}{b-a}+\frac{1}{b-c}=\frac{1}{a}+\frac{1}{c}$.

Answer 1

As, $b$ is the harmonic mean between $a$ and $c$

$\Rightarrow b=\frac{2ac}{a+c}$

We need to find $\frac{1}{b-a}+\frac{1}{b-c}=\frac{1}{\frac{2ac}{a+c}-a}+\frac{1}{\frac{2ac}{a+c}-c}$

$=\frac{a+c}{2ac-a(a+c)}+\frac{a+c}{2ac-c(a+c)}$

$=\frac{a+c}{2ac-a^2-ac}+\frac{a+c}{2ac-ca-c^2}$

$=\frac{a+c}{a(c-a)}-\frac{a+c}{c(c-a)}$

$=\frac{c(a+c)-a(a+c)}{ac(c-a)}$

$=\frac{c^2-a^2}{ac(c-a)}$

$=\frac{c+a}{ac}$

$=\frac{1}{a}+\frac{1}{c}$