The sum of the first two terms of a $G.P.$ is $1$ and every term of this series is twice its previous term. Then,the first term will be:

If the roots of the equation $3x^3-26x^2+52x-24=0$ are in geometric progression,then the sum of two of its roots is

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:

For $0 < \phi < \frac{\pi}{2}$,if $x = \sum_{n=0}^\infty \cos^{2n}\phi$,$y = \sum_{n=0}^\infty \sin^{2n}\phi$,and $z = \sum_{n=0}^\infty \cos^{2n}\phi \sin^{2n}\phi$,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?

The $G.M.$ of the numbers $3, 3^2, 3^3, ..., 3^n$ is