The first term of a $G.P.$ is $1$. The sum of the third term and fifth term is $90$. Find the common ratio of the $G.P.$

The first term of a $G.P.$ whose second term is $2$ and sum to infinity is $8$,will be

Three numbers are in $G.P.$ such that their sum is $38$ and their product is $1728$. The greatest number among them is

The terms of a $G.P.$ are positive. If each term is equal to the sum of the two terms that follow it,then the common ratio is:

The greatest positive integer $k,$ for which $49^k+1$ is a factor of the sum $49^{125}+49^{124}+\ldots+49^{2}+49+1$ is

Let $A_{1}, A_{2}, A_{3}, \ldots$ be squares such that for each $n \geq 1,$ the length of the side of $A_{n}$ equals the length of the diagonal of $A_{n+1}$. If the side length of $A_{1}$ is $12 \text{ cm}$,then the smallest value of $n$ for which the area of $A_{n}$ is less than $1 \text{ cm}^2$ is: