If $y = x - x^2 + x^3 - x^4 + \dots \infty$,then the value of $x$ is equal to:

If $x, y, z$ are in $G.P.$ and $a^x = b^y = c^z$,then

Let $G_1$ and $G_2$ be the geometric means of two series $x_1, x_2, \dots, x_n$ and $y_1, y_2, \dots, y_n$ respectively. If $G$ is the geometric mean of the series $\frac{x_i}{y_i}$ where $i = 1, 2, \dots, n$,then what is $G$ equal to?

If $s$ is the sum of an infinite $G.P.$ and $a$ is the first term,then the common ratio $r$ is given by:

If $|\alpha| < 1$ and $|\beta| < 1$,and $1 - \alpha + \alpha^2 - \alpha^3 + \dots \infty = s_1$ and $1 - \beta + \beta^2 - \beta^3 + \dots \infty = s_2$,then $1 - \alpha\beta + \alpha^2\beta^2 - \alpha^3\beta^3 + \dots \infty$ equals:

$A$ 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?