If $(20)^{19} + 2(21)(20)^{18} + 3(21)^2(20)^{17} + \ldots + 20(21)^{19} = k (20)^{19}$,then $k$ is equal to

If $u_{0}=8, u_{1}=3, u_{2}=12, u_{3}=51$,then the value of $\Delta^{3} u_{0}$ is

If $S = \frac{7}{5} + \frac{9}{5^{2}} + \frac{13}{5^{3}} + \frac{19}{5^{4}} + \ldots$, then $160 \,S$ is equal to....... .

For $k \in N$,if the sum of the series $1+\frac{4}{k}+\frac{8}{k^2}+\frac{13}{k^3}+\frac{19}{k^4}+\ldots$ is $10$,then the value of $k$ is

The sum of $1 + \frac{2}{5} + \frac{3}{5^2} + \frac{4}{5^3} + \dots$ up to $n$ terms is

Let $f(x) = \sin x + 2\sin^2 x + 3\sin^3 x + 4\sin^4 x + \dots \infty$. Then the number of solutions of the equation $f(x) = 2$ in $x \in [-\pi, \pi] - \{\pm \frac{\pi}{2}\}$ is