If the roots of $x^3-42x^2+336x-512=0$ are in increasing geometric progression,then its common ratio is: (in $:1$)

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

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 -

Show that the ratio of the sum of the first $n$ terms of a $G.P.$ to the sum of the terms from the $(n+1)^{th}$ to the $(2n)^{th}$ term is $\frac{1}{r^{n}}$.

If $r > 1$,$x = a + \frac{a}{r} + \frac{a}{r^2} + \dots \infty$,$y = b - \frac{b}{r} + \frac{b}{r^2} - \dots \infty$,and $z = c + \frac{c}{r^2} + \frac{c}{r^4} + \dots \infty$,then $\frac{xy}{z} = \dots$

Let $a_1, a_2, ..., a_{10}$ be a $G.P.$ If $\frac{a_3}{a_1} = 25$,then $\frac{a_9}{a_5}$ is equal to: