The sum of $n$ terms of the following series $1 + (1 + x) + (1 + x + x^2) + \dots$ will be

Let $a$ and $b$ be roots of $x^2 - 3x + p = 0$ and let $c$ and $d$ be the roots of $x^2 - 12x + q = 0$,where $a, b, c, d$ form an increasing $G$.$P$. Then the ratio of $(q + p) : (q - p)$ is equal to

Let ${S_n}$ denote the sum of $n$ terms of an $A.P.$ If ${S_{2n}} = 3{S_n}$,then the ratio $\frac{{{S_{3n}}}}{{{S_n}}}$ is equal to:

If $\log_3 2, \log_3(2^x - 5)$ and $\log_3(2^x - 7/2)$ are in $A.P.$,then $x$ is equal to

The sum of $100$ terms of the series $0.9 + 0.09 + 0.009 + \dots$ will be

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