Let $C_0$ be a circle of radius $1$. For $n \geq 1$,let $C_n$ be a circle whose area equals the area of a square inscribed in $C_{n-1}$. Then,$\sum_{i=0}^{\infty} \text{Area}(C_i)$ equals

In an increasing geometric series,the sum of the second and the sixth term is $\frac{25}{2}$ and the product of the third and fifth term is $25$. Then,the sum of the $4^{\text{th}}$,$6^{\text{th}}$,and $8^{\text{th}}$ terms is equal to

$A$ $G.P.$ consists of an even number of terms. If the sum of all the terms is $5$ times the sum of terms occupying odd places,then find its common ratio.

$A$ 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 $\frac{6}{3^{12}} + \frac{10}{3^{11}} + \frac{20}{3^{10}} + \frac{40}{3^{9}} + \dots + \frac{10240}{3} = 2^{n} \cdot m$,where $m$ is odd,then $m \cdot n$ is equal to

If $x, 2x + 2$,and $3x + 3$ are in a geometric progression,what is the fourth term?