Question

Given $a,b,c$ are in $A.P.$, $b,c,d$ are in $G.P.$ and $c,d,e$ are in $H.P.$ If $a=2$ and $e=18$, then the sum of all possible value of $‘c^{\prime }$ is

Answer 1

$a,b,c$ are in $A.P.$,
$b=\frac{a+c}{2}$
$b,c,d$ are in $G.P.$,
$c^2=bd$
$c,d,e$ are in $H.P.$,
$d=\frac{2ce}{c+e}$
So,
$c^2=\left(\frac{a+c}{2}\right)\times \left(\frac{2ce}{c+e}\right)\Rightarrow c^2+ce=ae+ce\Rightarrow c^2=ae=2\times 18\Rightarrow c=\pm 6$

Sum of all value will be $0$.