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

If $p,\;q,\;r$ are in one geometric progression and $a,\;b,\;c$ in another geometric progression, then $cp,\;bq,\;ar$ are in

Insert two numbers between $3$ and $81$ so that the resulting sequence is $G.P.$

If the first and the $n^{\text {th }}$ term of a $G.P.$ are $a$ and $b$, respectively, and if $P$ is the product of $n$ terms, prove that $P^{2}=(a b)^{n}$

Which term of the following sequences:

$\quad 2,2 \sqrt{2}, 4, \ldots$ is $128 ?$

The value of ${a^{{{\log }_b}x}}$, where $a = 0.2,\;b = \sqrt 5 ,\;x = \frac{1}{4} + \frac{1}{8} + \frac{1}{{16}} + .........$to $\infty $ is