Question

If $2,b,c$ are in A.P., $b,c,d$ are in G.P., and $c,d,18$ are in H.P. $($where $b,c,d\in R^{+})$, then the value of $b+c+d$ is

Answer 1

Here, $2b=2+c,c^2=bd,d=\frac{2\left(18\right)\left(c\right)}{c+18}$
Now $c^2=bd=\frac{2+c}{2}\times \frac{2\left(18\right)\left(c\right)}{c+18}$
$\Rightarrow c=6(\because c\in R^{+})$
$\Rightarrow 2b=8\Rightarrow b=4$
$\Rightarrow d=\frac{c^2}{b}=9$
$\therefore b+c+d=19$