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 the positive numbers $a _1, a _2, a _3, a _4$ and $a _5$ be in a G.P. Let their mean and variance be $\frac{31}{10}$ and $\frac{ m }{ n }$ respectively, where $m$ and $n$ are co-prime. If the mean of their reciprocals is $\frac{31}{40}$ and $a_3+a_4+a_5=14$, then $m + n$ is equal to $.........$.

If the ${10^{th}}$ term of a geometric progression is $9$ and ${4^{th}}$ term is $4$, then its ${7^{th}}$ term is

Let $a _1, a _2, a _3, \ldots$ be a $G.P.$ of increasing positive numbers. Let the sum of its $6^{\text {th }}$ and $8^{\text {th }}$ terms be $2$ and the product of its $3^{\text {rd }}$ and $5^{\text {th }}$ terms be $\frac{1}{9}$. Then $6\left( a _2+\right.$ $\left.a_4\right)\left(a_4+a_6\right)$ is equal to

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 $a_1, a_2, a_3, \ldots$. be a $GP$ of increasing positive numbers. If the product of fourth and sixth terms is $9$ and the sum of fifth and seventh terms is $24 ,$ then $a_1 a_9+a_2 a_4 a_9+a_5+a_7$ is equal to $.........$.