If we insert two numbers between $\sqrt{2}$ and $4$ so that the resulting sequence is in $G.P.$, then the inserted numbers in the order are

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

Find the Geometric Mean $(G.M.)$ of the sequence $1, 2, 2^2, \dots, 2^n$.

The geometric series $a + ar + ar^2 + ar^3 + \dots \infty$ has a sum of $7$,and the sum of the terms involving odd powers of $r$ is $3$. Then the value of $(a^2 - r^2)$ is -

Let $A_{1}, A_{2}, A_{3}, \ldots$ be an increasing geometric progression of positive real numbers. If $A_{1} A_{3} A_{5} A_{7} = \frac{1}{1296}$ and $A_{2} + A_{4} = \frac{7}{36}$,then the value of $A_{6} + A_{8} + A_{10}$ is equal to

If the $4^{th}$,$7^{th}$,and $10^{th}$ terms of a Geometric Progression $(GP)$ are $a, b,$ and $c$ respectively,then: