If the sum and product of four positive consecutive terms of a $G.P.$, are $126$ and $1296$, respectively, then the sum of common ratios of all such $GPs$ is $………$.

The sum of few terms of any ratio series is $728$, if common ratio is $3$ and last term is $486$, then first term of series will be

The ${4^{th}}$ term of a $G.P.$ is square of its second term, and the first term is $-3$ Determine its $7^{\text {th }}$ term.

If  ${x_r} = \cos (\pi /{3^r}) – i\sin (\pi /{3^r}),$ (where  $i = \sqrt{-1}),$ then value  of $x_1.x_2.x_3……\infty ,$ 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