Question

Prove that
$\frac{a+b+c}{a^{-1}{b}^{-1}+b^{-1}{c}^{-1}+c^{-1}{a}^{-1}}=abc$

Answer 1

Consider,

$\frac{a+b+c}{a^{-1}{b}^{-1}+b^{-1}{c}^{-1}+c^{-1}{a}^{-1}}$

$=\frac{a+b+c}{\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca}}$-----since $x^{-1}=\frac{1}{x}$

$=\frac{a+b+c}{\frac{c}{abc}+\frac{a}{abc}+\frac{b}{cab}}$ -----Multiply and divide each terms in the denominator by the missing terms. i.e., $c,a,b$ respectively

$=\frac{a+b+c}{\frac{c+a+b}{abc}}$

$=\frac{1}{\frac{1}{abc}}$

$=abc$

Hence proved.