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$
$\Rightarrow \frac{b-ab^{2}-a-b}{1-ab} =\frac{ b+c-b+b^{2}c}{1-bc}$
$ \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 $H.P$.