If $f(x)$ is a function satisfying $f(x + y) = f(x)f(y)$ for all $x, y \in N$ such that $f(1) = 3$ and $\sum_{x = 1}^n f(x) = 120$,then the value of $n$ is

The sum $\frac{3 \times 1}{1^2} + \frac{5 \times (1^3 + 2^3)}{1^2 + 2^2} + \frac{7 \times (1^3 + 2^3 + 3^3)}{1^2 + 2^2 + 3^2} + \dots$ up to the $10^{th}$ term 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

If $m$ arithmetic means $(A.Ms)$ and three geometric means $(G.Ms)$ are inserted between $3$ and $243$ such that the $4^{\text{th}}$ $A.M.$ is equal to the $2^{\text{nd}}$ $G.M.$,then $m$ is equal to:

If the sum of three consecutive terms of an $A.P.$ is $51$ and the product of the last and first term is $273$,then the numbers are

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: