If the sum of infinite terms of a $G.P.$ is $3$ and the sum of the squares of its terms is $3$,then its first term and common ratio are:

The third term of a $G$.$P$. is $9$. The product of its first five terms is

For the functions $f(\theta) = \alpha \tan^2 \theta + \beta \cot^2 \theta$ and $g(\theta) = \alpha \sin^2 \theta + \beta \cos^2 \theta$, where $\alpha > \beta > 0$, let $\min_{0 < \theta < \pi/2} f(\theta) = \max_{0 < \theta < \pi} g(\theta)$. If the first term of a $G$.$P$. is $(\frac{\alpha}{2\beta})$, its common ratio is $(\frac{2\beta}{\alpha})$ and the sum of its first $10$ terms is $\frac{m}{n}$, where $\gcd(m,n)=1$, then $m+n$ is equal to . . . . . . .

Insert three numbers between $1$ and $256$ so that the resulting sequence is a $G.P.$

If $G_1$ and $G_2$ are two geometric means between two numbers and $A$ is the arithmetic mean between them,then find the value of $\frac{G_1^2}{G_2} + \frac{G_2^2}{G_1}$.

$A$ person writes a letter to four of his friends. He asks each one of them to copy the letter and mail it to four different persons with the instruction that they continue the chain similarly. Assuming that the chain is not broken and that it costs $50$ paise to mail one letter,find the total amount spent on postage when the $8^{\text{th}}$ set of letters is mailed.