are in A.P., prove that a, b, c are in A.P.

Given;

$a\left(\frac{1}{b}+\frac{1}{c}\right), b\left(\frac{1}{c}+\frac{1}{a}\right), c\left(\frac{1}{a}+\frac{1}{b}\right)$ are in A.P.

By adding 1 to each term, we get:

$a\left(\frac{1}{b}+\frac{1}{c}\right)+1, b\left(\frac{1}{c}+\frac{1}{a}\right)+1, c\left(\frac{1}{a}+\frac{1}{b}\right)+1$ are in A.P.

$\Rightarrow a\left(\frac{1}{b}+\frac{1}{c}+\frac{1}{a}\right), b\left(\frac{1}{c}+\frac{1}{a}+\frac{1}{b}\right), c\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)$ are in A.P.

Dividing all terms by $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$, we get:

$\Rightarrow a, b, c$ are in A.P.

Hence proved.