If $G$ is the geometric mean between $x$ and $y$,then the value of $\frac{1}{G^2 - x^2} + \frac{1}{G^2 - y^2}$ is:

The roots of the equation $x^5 - 40x^4 + px^3 + qx^2 + rx + s = 0$ are in $G.P.$ The sum of their reciprocals is $10$. Then the value of $|s|$ is

$A$ 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.

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:

The sum of $100$ terms of the series $0.9 + 0.09 + 0.009 + \dots$ will be

If the first term of an infinite geometric series is twice the sum of all its subsequent terms,what is the common ratio?