If $\frac{a + bx}{a - bx} = \frac{b + cx}{b - cx} = \frac{c + dx}{c - dx}$ $(x \ne 0)$,then $a, b, c, d$ are in

In a geometric progression consisting of positive terms,each term equals the sum of the next two terms. Then the common ratio of its progression is equal to:

Find a $G.P.$ for which the sum of the first two terms is $-4$ and the fifth term is $4$ times the third term.

The arithmetic mean of two numbers $b$ and $c$ is $a$,and $g_1$ and $g_2$ are two geometric means between them. If $g_1^3 + g_2^3 = kabc$,then $k = \dots$

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

If $a = \sum_{n=0}^\infty x^n$,$b = \sum_{n=0}^\infty y^n$,and $c = \sum_{n=0}^\infty (xy)^n$,where $|x|, |y| < 1$,then which of the following is true?