An algebra problem by Rishik Jain

Algebra Level 2

If $abc=1$ , find

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

1 solution

Sajjad Sajjad
Mar 23, 2015

By simplifying the given form we get,

$\frac{b}{b + ab + 1} + \frac{c}{c + bc +1} \frac{a}{a + ca + 1}$

As, $abc = 1$

$ab = \frac{1}{c}$ and $a = \frac{1}{bc}$

Puting the value we get,

$\frac{bc}{bc + c + 1} + \frac{c}{bc + c + 1} + \frac{1}{bc + c + 1}$

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

$= 1$ :)