The third term of a $G.P.$ is the square of the first term. If the second term is $8$,then the $6^{th}$ term is:

What is the product of $n$ geometric means between $a$ and $b$?

The difference between the fourth term and the first term of a Geometric Progression is $52$. If the sum of its first three terms is $26$,then the sum of the first six terms of the progression is

Show that the products of the corresponding terms of the sequences $a, ar, ar^{2}, \dots, ar^{n-1}$ and $A, AR, AR^{2}, \dots, AR^{n-1}$ form a $G.P.$,and find the common ratio.

Let ${a_n}$ be the ${n^{th}}$ term of a $G$.$P$. 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

Find the sum of the products of the corresponding terms of the sequences $2, 4, 8, 16, 32$ and $128, 32, 8, 2, \frac{1}{2}$.