The sum of $3$ numbers in geometric progression is $38$ and their product is $1728$. The middle number is

If the $p^{\text {th }}, q^{\text {th }}$ and $r^{\text {th }}$ terms of a $G.P.$ are $a, b$ and $c,$ respectively. Prove that

$a^{q-r} b^{r-p} c^{p-q}=1$

The sum of an infinite geometric series with positive terms is $3$ and the sum of the cubes of its terms is $\frac {27}{19}$. Then the common ratio of this series is

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

Let the first term $a$ and the common ratio $r$ of a geometric progression be positive integers. If the sum of its squares of first three terms is $33033$, then the sum of these three terms is equal to

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