Question

If $\frac{a+b}{1-ab},b,\frac{b+c}{1-bc}$ be in A.P., then prove that $a,b^{-1}$, c are in H.P.

Answer 1

We have to prove that
$b^{-1}=\frac{2ac}{a+c}$
We are given that
$b-\frac{a+b}{1-ab}=\frac{b+,c}{1-bc}-b$
$or\frac{-a(b^2+1)}{1-ab}=\frac{c(1+b^2)}{1-bc}$
or $-a+abc=c-abcora+c=2abc$

Hence proved