If $\frac{6}{3^{12}}+\frac{10}{3^{11}}+\frac{20}{3^{10}}+\frac{40}{3^{9}}+\ldots . .+\frac{10240}{3}=2^{ n } \cdot m$, where $m$ is odd, then $m . n$ is equal to

The first two terms of a geometric progression add up to $12.$ the sum of the third and the fourth terms is $48.$ If the terms of the geometric progression are alternately positive and negative, then the first term is

The value of $\overline {0.037} $ where,  $\overline {.037} $ stands for the number $0.037037037........$ is

If ${(p + q)^{th}}$ term of a $G.P.$ be $m$ and ${(p - q)^{th}}$ term be $n$, then the ${p^{th}}$ term will be

Find the sum to $n$ terms of the sequence, $8,88,888,8888 \ldots$

Suppose the sides of a triangle form a geometric progression with common ratio $r$. Then, $r$ lies in the interval