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

The sum of the first $20$ terms of the sequence $0.7, 0.77, 0.777, \dots$ is

If $n$ is the smallest natural number such that $n+2n+3n+\ldots+99n$ is a perfect square,then the number of digits of $n^2$ is

The sum to $20$ terms of the series $2^2-3^2+4^2-5^2+6^2-\ldots$ is equal to $........$.

If $A = \sum_{n=1}^{\infty} \frac{1}{(3+(-1)^{n})^{n}}$ and $B = \sum_{n=1}^{\infty} \frac{(-1)^{n}}{(3+(-1)^{n})^{n}}$,then $\frac{A}{B}$ is equal to:

For $x \in \mathbb{R}$,let $[x]$ denote the greatest integer $\le x$. Find the sum of the series $\left[ -\frac{1}{3} \right] + \left[ -\frac{1}{3} - \frac{1}{100} \right] + \left[ -\frac{1}{3} - \frac{2}{100} \right] + \dots + \left[ -\frac{1}{3} - \frac{99}{100} \right]$.