Show that the products of the corresponding terms of the sequences $a,$ $ar,$ $a r^{2},$ $……a r^{n-1}$ and $A, A R, A R^{2}, \ldots, A R^{n-1}$ form a $G .P.,$ and find the common ratio.

The sum of first two terms of a $G.P.$ is $1$ and every term of this series is twice of its previous term, then the first term will be

If $a$,$b$,$c \in {R^ + }$ are such that $2a$,$b$ and $4c$ are in $A$.$P$ and $c$,$a$ and $b$ are in $G$.$P$., then

If the first term of a $G.P. a_1, a_2, a_3……$ is unity such that $4a_2 + 5a_3$ is least, then the common ratio of $G.P.$ is

Let for $n =1,2, \ldots \ldots, 50, S _{ a }$ be the sum of the infinite geometric progression whose first term is $n ^{2}$ and whose common ratio is $\frac{1}{(n+1)^{2}}$. Then the value of $\frac{1}{26}+\sum\limits_{n=1}^{50}\left(S_{n}+\frac{2}{n+1}-n-1\right)$ is equal to