If $p,\;q,\;r$ are in one geometric progression and $a,\;b,\;c$ in another geometric progression, then $cp,\;bq,\;ar$ are in

If $s$ is the sum of an infinite $G.P.$, the first term $a$ then the common ratio $r$ given by

Let ${A_n} = \left( {\frac{3}{4}} \right) - {\left( {\frac{3}{4}} \right)^2} + {\left( {\frac{3}{4}} \right)^3} - ..... + {\left( { - 1} \right)^{n - 1}}{\left( {\frac{3}{4}} \right)^n}$ and $B_n \,= 1 - A_n$ . Then, the least odd natural number $p$ , so that ${B_n} > {A_n}$, for all $n \geq p$ is

In an increasing geometric progression ol positive terms, the sum of the second and sixth terms is $\frac{70}{3}$ and the product of the third and fifth terms is $49$. Then the sum of the $4^{\text {th }}, 6^{\text {th }}$ and $8^{\text {th }}$ terms is :-

The sum of first three terms of a $G.P.$ is $16$ and the sum of the next three terms is
$128.$ Determine the first term, the common ratio and the sum to $n$ terms of the $G.P.$

The interior angle of a $'n$' sided convex polygon are in $G.P$.. The smallest angle is $1^o $ and common ratio is $2^o $ then number of possible values of $'n'$ is