Question

If $a,b,c$ are in H.P., $b,c,d$ are in $G.P$. and $c,d,e$ are in $A.P$., then the value of $e$ is

• A. $\frac{a{b}^2}{(2a-b{)}^2}$
• B. $\frac{a^{2}b}{(2a-b{)}^2}$
• C. $\frac{a^2{b}^2}{(2a-b{)}^2}$
• D. None of these

Answer 1

Answer: (A)

$a,b,c$ are in $H.P.\Rightarrow b=\frac{2ac}{a+c}(i)$
$b,c,d$ are in $G.P.\Rightarrow c^2=bd(ii)$
$c,d,e$ are in $A.P.\Rightarrow d=\frac{c+e}{2}(iii)$
From (i), $ab+bc=2ac$
$\Rightarrow c=\frac{ab}{2a-b}(iv)$
From (iii) and (iv),
$d=\frac{1}{2}\left[\frac{ab}{2a-b}+e\right](v)$
From (ii), (iv) and (v),
$\frac{a^2{b}^2}{(2a-b{)}^2}=\frac{b}{2}\left[\frac{ab}{2a-b}+e\right]$
$\Rightarrow e=\frac{a{b}^2}{(2a-b{)}^2}$