Fifth term of a $G.P.$ is $2$, then the product of its $9$ terms is

An $A.P.$, a $G.P.$ and a $H.P.$ have the same first and last terms and the same odd number of terms. The middle terms of the three series are in

Which term of the following sequences:

$\sqrt{3}, 3,3 \sqrt{3}, \ldots$ is $729 ?$

If $a _{1}(>0), a _{2}, a _{3}, a _{4}, a _{5}$ are in a G.P., $a _{2}+ a _{4}=2 a _{3}+1$ and $3 a _{2}+ a _{3}=2 a _{4}$, then $a _{2}+ a _{4}+2 a _{5}$ is equal to

Let $a, b, c, d$ and $p$ be any non zero distinct real numbers such that  $\left(a^{2}+b^{2}+c^{2}\right) p^{2}-2(a b+b c+ cd ) p +\left( b ^{2}+ c ^{2}+ d ^{2}\right)=0 .$ Then

The sum of the first $n$ terms of the series $\frac{1}{2} + \frac{3}{4} + \frac{7}{8} + \frac{{15}}{{16}} + .........$ is