If the first term of a $G.P.$ ${a_1},\;{a_2},\;{a_3},..........$ is unity such that $4{a_2} + 5{a_3}$ is least, then the common ratio of $G.P.$ is

The sum of some terms of $G.P.$ is $315$ whose first term and the common ratio are $5$ and $2,$ respectively. Find the last term and the number of terms.

For $0<\mathrm{c}<\mathrm{b}<\mathrm{a}$, let $(\mathrm{a}+\mathrm{b}-2 \mathrm{c}) \mathrm{x}^2+(\mathrm{b}+\mathrm{c}-2 \mathrm{a}) \mathrm{x}$ $+(c+a-2 b)=0$ and $\alpha \neq 1$ be one of its root. Then, among the two statements

$(I)$ If $\alpha \in(-1,0)$, then $\mathrm{b}$ cannot be the geometric mean of $\mathrm{a}$ and $\mathrm{c}$

$(II)$ If $\alpha \in(0,1)$, then $\mathrm{b}$ may be the geometric mean of $a$ and $c$

A $G.P.$ consists of an even number of terms. If the sum of all the terms is $5$ times the sum of terms occupying odd places, then find its common ratio.

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

If the sum of an infinite $G.P.$ and the sum of square of its terms is $3$, then the common ratio of the first series is