If $x, 2x + 2, 3x + 3$ are in $G.P.$,then the fourth term is

If $2^{10} + 2^{9} \cdot 3^{1} + 2^{8} \cdot 3^{2} + \ldots + 2^{1} \cdot 3^{9} + 3^{10} = S - 2^{11}$,then $S$ is equal to

If the first and the $n^{\text{th}}$ term of a $G.P.$ are $a$ and $b$ respectively,and if $P$ is the product of $n$ terms,prove that $P^{2} = (ab)^{n}$.

The sum of the series $5.05 + 1.212 + 0.29088 + \dots \infty$ is

The sum of some terms of a $G.P.$ is $315$,whose first term and the common ratio are $5$ and $2$ respectively. Find the last term and the number of terms.

If $S_n$ stands for the sum of $n$ terms of a $GP$ with $a$ as the first term and $r$ as the common ratio,then $S_n : S_{2n}$ is