Question

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

• A. A.P.
• B. G.P.
• C. H.P.
• D. none of these

Answer 1

Answer: (C)

By the given condition
$b-\frac{a+b}{1-ab}=\frac{b+c}{1-ab}-b$
$\Rightarrow \frac{b-a{b}^2-a-b}{1-ab}=\frac{b+c-b+b^{2}c}{1-bc}$
$\Rightarrow \frac{-a\left(1+b^2\right)}{1-ab}=\frac{c\left(1+b^2\right)}{1-bc}$
$\Rightarrow \frac{-a}{1-ab}=\frac{c}{1-bc}$
$\Rightarrow -a+abc=c-abc$
$\Rightarrow 2abc=c+a$
$\Rightarrow \frac{1}{b}=\frac{2ac}{a+c}$
$\Rightarrow a,\frac{1}{b},c$ are in $H.P$.