If $[x]$ denotes the greatest integer less than or equal to $x$,then $\sum_{n=8}^{100} \left[ \frac{(-1)^{n} n}{2} \right]$ 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:

If $\alpha, \beta$ are natural numbers such that $100^{\alpha} - 199\beta = (100)(100) + (99)(101) + (98)(102) + \ldots + (1)(199)$,then the slope of the line passing through $(\alpha, \beta)$ and the origin is:

The sum of the infinite series $1+\frac{5}{6}+\frac{12}{6^{2}}+\frac{22}{6^{3}}+\frac{35}{6^{4}}+\frac{51}{6^{5}}+\frac{70}{6^{6}}+\ldots$ is equal to

$2 + 4 + 7 + 11 + 16 + \dots$ to $n$ terms =

The sum of infinite terms of the geometric progression $\frac{\sqrt{2} + 1}{\sqrt{2} - 1}, \frac{1}{2 - \sqrt{2}}, \frac{1}{2}, \dots$ is