Let $a, a r, a r^2, \ldots . . .$. be an infinite $G.P.$ If $\sum_{n=0}^{\infty} a^n=57$ and $\sum_{n=0}^{\infty} a^3 r^{3 n}=9747$, then $a+18 r$ is equal to :
$27$
$46$
$38$
$31$
A person writes a letter to four of his friends. He asks each one of them to copy the letter and mail to four different persons with instruction that they move the chain similarly. Assuming that the chain is not broken and that it costs $50$ paise to mail one letter. Find the amount spent on the postage when $8^{\text {th }}$ set of letter is mailed.
If $\frac{{x + y}}{2},\;y,\;\frac{{y + z}}{2}$ are in $H.P.$, then $x,\;y,\;z$ are in
Find the sum of $n$ terms in the geometric progression $\sqrt{7}, \sqrt{21}, 3 \sqrt{7}, \ldots$
If the third term of a $G.P.$ is $4$ then the product of its first $5$ terms is
Let $C_0$ be a circle of radius $I$ . For $n \geq 1$, let $C_n$ be a circle whose area equals the area of a square inscribed in $C_{n-1} .$ Then, $\sum \limits_{i=0}^{\infty}$ Area $\left(C_i\right)$ equals