$\sum_{k=1}^{\infty} \sum_{r=0}^k \frac{1}{3^k} \binom{k}{r}$ is equal to

Let $S_k = \frac{1 + 2 + 3 + .... + k}{k}$. If $S_1^2 + S_2^2 + ....... + S_{10}^2 = \frac{5}{12}A$,then $A$ is equal to:

$\sum_{k=1}^{2n+1} (-1)^{k-1} \cdot k^2$ is equal to

What is the sum of the series $1 + (1 + x) + (1 + x + x^2) + (1 + x + x^2 + x^3) + \dots$ up to $n$ terms?

The value of $\overline{0.037}$,where $\overline{0.037}$ stands for the number $0.037037037...$,is

Let $75 \ldots 57$ denote the $(r+2)$ digit number where the first and the last digits are $7$ and the remaining $r$ digits are $5$. Consider the sum $S = 77 + 757 + 7557 + \ldots + 75 \ldots 57$ (where the last term has $98$ digits). If $S = \frac{75 \ldots 57 + m}{n}$,where $m$ and $n$ are natural numbers less than $3000$,then the value of $m + n$ is: