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

If $x = \sum\limits_{n = 0}^\infty {{a^n}} ,\;y = \sum\limits_{n = 0}^\infty {{b^n},\;z = \sum\limits_{n = 0}^\infty {{{(ab)}^n}} } $,where $a, b < 1$,then

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

In a $GP$ series consisting of positive terms, each term is equal to the sum of the next two terms. Then, the common ratio of this $GP$ series is

$A$ geometric progression consists of positive terms. If each term is equal to the sum of the next two terms,what is the common ratio of the progression?

If each term of a geometric progression $a_1, a_2, a_3, \ldots$ with $a_1 = \frac{1}{8}$ and $a_2 \neq a_1$ is the arithmetic mean of the next two terms and $S_n = a_1 + a_2 + \ldots + a_n$,then $S_{20} - S_{18}$ is equal to