The G.M. of the numbers $3,\,{3^2},\,{3^3},\,......,\,{3^n}$ is

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

If the first term of a $G.P.$ be $5$ and common ratio be $ - 5$, then which term is $3125$

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 $.........$.

Find the sum to indicated number of terms in each of the geometric progressions in $\left.1,-a, a^{2},-a^{3}, \ldots n \text { terms (if } a \neq-1\right)$

If $2(y - a)$ is the $H.M.$ between $y - x$ and $y - z$, then $x - a,\;y - a,\;z - a$ are in