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

Let $n \geq 3$ and let $C_1, C_2, \ldots, C_n$, be circles with radii $r_1, r_2, \ldots, r_n$, respectively. Assume that $C_i$ and $C_{i+1}$ touch externally for $1 \leq i \leq n-1$. It is also given that the $X$-axis and the line $y=2 \sqrt{2} x+10$ are tangential to each of the circles. Then, $r_1, r_2, \ldots, r_n$ are in

Let ${a_1},{a_2}...,{a_{10}}$ be a $G.P.$ If $\frac{{{a_3}}}{{{a_1}}} = 25,$ then $\frac {{{a_9}}}{{{a_{  5}}}}$ equal

The terms of a $G.P.$ are positive. If each term is equal to the sum of two terms that follow it, then the common ratio is

If $a,\;b,\;c$ are in $G.P.$, then

The $4^{\text {tht }}$ term of $GP$ is $500$ and its common ratio is $\frac{1}{m}, m \in N$. Let $S_n$ denote the sum of the first $n$ terms of this GP. If $S_6 > S_5+1$ and $S_7 < S_6+\frac{1}{2}$, then the number of possible values of $m$ is $..........$