If the ratio of the sum of the first three terms and the sum of the first six terms of a $G.P.$ is $125 : 152$,then the common ratio $r$ is:

The roots of the equation $x^3-14x^2+56x-64=0$ are in

What is the sum of the first $100$ terms of the series $0.9 + 0.09 + 0.009 + \dots$?

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

$A$ particle starts at the origin and moves $1$ unit horizontally to the right and reaches $P_{1}$, then it moves $\frac{1}{2}$ unit vertically up and reaches $P_{2}$, then it moves $\frac{1}{4}$ unit horizontally to the right and reaches $P_{3}$, then it moves $\frac{1}{8}$ unit vertically down and reaches $P_{4}$, then it moves $\frac{1}{16}$ unit horizontally to the right and reaches $P_{5}$ and so on. Let $P_{n} = (x_{n}, y_{n})$ and $\lim_{n \rightarrow \infty} x_{n} = \alpha$ and $\lim_{n \rightarrow \infty} y_{n} = \beta$. Then, $(\alpha, \beta)$ is

If $1 + r + r^2 + \dots + r^n = (1 + r) (1 + r^2) (1 + r^4) (1 + r^8)$,then what is the value of $n$?