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

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:

Let $\{a_{n}\}_{n=0}^{\infty}$ be a sequence such that $a_{0}=a_{1}=0$ and $a_{n+2}=2a_{n+1}-a_{n}+1$ for all $n \geq 0$. Then,$\sum\limits_{n=2}^{\infty} \frac{a_{n}}{7^{n}}$ is equal to

The $n^{th}$ term of the series $2 + 4 + 7 + 11 + \dots$ is

The mean of $10$ numbers $7 \times 8, 10 \times 10, 13 \times 12, 16 \times 14, \ldots$ is ....... .

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