The number of solutions of $2^{1+|\cos x|+|\cos x|^2+\ldots} = 4$ in $(-\pi, \pi)$ is

The sum of the series,$\frac{1}{2 \cdot 3} \cdot 2 + \frac{2}{3 \cdot 4} \cdot 2^{2} + \frac{3}{4 \cdot 5} \cdot 2^{3} + \ldots$ up to $n$ terms is

Let $\alpha = \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \dots \infty$ and $\beta = \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \dots \infty$. Then the value of $(0.2)^{\log_{\sqrt{5}}(\alpha)} + (0.04)^{\log_{5}(\beta)}$ is equal to:

Find the sum of the following series up to $n$ terms:
$5+55+555+\ldots$

The sum of $n$ terms of the following series $1 + (1 + x) + (1 + x + x^2) + \dots$ will be

If $1 \cdot 3 \cdot 5 + 3 \cdot 5 \cdot 7 + 5 \cdot 7 \cdot 9 + \ldots \text{ to } n \text{ terms} = n(n+1) f(n) - 3n$,then $f(1) =$