If the sum of an infinite $G.P.$ and the sum of the squares of its terms is $3$,then the common ratio of the first series is

The sum of the first ten terms of a geometric progression is $S_1$ and the sum of the next ten terms ($11^{th}$ to $20^{th}$) is $S_2$. What is the 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

For a sequence,$a_1 = 2$ and $\frac{a_{n+1}}{a_n} = \frac{1}{3}$. Then $\sum_{r=1}^{20} a_r$ is

The geometric mean of $n$ positive terms $x_1, x_2, \dots, x_n$ is equal to:

For a sequence $(t_{n})$,if $s_{n} = 7(3^{n} - 1)$,then $t_{n} =$