If the third term of a $G.P.$ is $4$ then the product of its first $5$ terms is

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 infinite terms of the geometric progression $\frac{{\sqrt 2 + 1}}{{\sqrt 2 – 1}},\frac{1}{{2 – \sqrt 2 }},\frac{1}{2}…..$ is

Let $b_1, b_2,……, b_n$ be a geometric sequence such that $b_1 + b_2 = 1$ and $\sum\limits_{k = 1}^\infty  {{b_k} = 2} $ Given that $b_2 < 0$ , then the value of $b_1$ is 

Let ${a_n}$ be the ${n^{th}}$ term of the G.P. of positive numbers. Let $\sum\limits_{n = 1}^{100} {{a_{2n}}} = \alpha $ and $\sum\limits_{n = 1}^{100} {{a_{2n – 1}}} = \beta $, such that $\alpha \ne \beta $,then the common ratio is