$A$ $G.P.$ consists of an even number of terms. If the sum of all the terms is $5$ times the sum of the terms occupying odd places,then the common ratio will be equal to

Let $S_1, S_2, \dots$ be squares such that for each $n \ge 1$,the length of a side of $S_n$ equals the length of a diagonal of $S_{n+1}$. If the length of a side of $S_1$ is $10 \ cm$,then for which of the following values of $n$ is the area of $S_n$ less than $1 \ cm^2$?

The numbers $(\sqrt{2} + 1), 1, (\sqrt{2} - 1)$ will be in

If the sum of the second,fourth and sixth terms of a $G.P.$ of positive terms is $21$ and the sum of its eighth,tenth and twelfth terms is $15309$,then the sum of its first nine terms is:

The number of ways of selecting $3$ numbers that are in $G.P.$ from the set $\{1, 2, 3, \ldots, 100\}$ is

Let $a, ar, ar^2, \ldots$ be an infinite $G.P.$ If $\sum_{n=0}^{\infty} ar^n = 57$ and $\sum_{n=0}^{\infty} a^3 r^{3n} = 9747$,then $a + 18r$ is equal to: