If the sum of the first $6$ terms is $9$ times the sum of the first $3$ terms of the same $G.P.$,then the common ratio of the series will be

The sum of an infinite geometric series is $3$. $A$ series,which is formed by squares of its terms,has the sum also $3$. The first series is

If $\log 2, \log (2^n - 1)$ and $\log (2^n + 3)$ are in $A.P.$,then $n =$

If the arithmetic mean of two numbers $a$ and $b$,$a > b > 0$,is five times their geometric mean,then $\frac{a + b}{a - b}$ is equal to

If $a$ is the arithmetic mean of $b$ and $c$,and $G_1, G_2$ are the two geometric means between them,then $G_1^3 + G_2^3 = $

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