Question

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

Answer 1

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

LHS:

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

$\Rightarrow \frac{a+b+c}{\frac{1}{a}\times \frac{1}{b}+\frac{1}{b}\times \frac{1}{c}+\frac{1}{a}\times \frac{1}{c}}$

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

$\Rightarrow \frac{a+b+c}{1}\times \frac{abc}{a+b+c}$

$\Rightarrow abc=$RHS proved.