Find the sum $S_n = 1 + 2x + 3x^2 + 4x^3 + \dots$ up to $n$ terms.

Evaluate the sum: $\sum\limits_{k = 1}^n {k\left( {1 + \frac{1}{n}} \right)^{k - 1}}$

Suppose $a_1, a_2, 2, a_3, a_4$ are in an arithmetico-geometric progression. If the common ratio of the corresponding geometric progression is $2$ and the sum of all $5$ terms of the arithmetico-geometric progression is $\frac{49}{2}$,then $a_4$ is equal to $...........$.

Let $S = 109 + \frac{108}{5} + \frac{107}{5^2} + \ldots + \frac{2}{5^{107}} + \frac{1}{5^{108}}$. Then the value of $(16S - (25)^{-54})$ is equal to $............$.

If the square root of $a^{\frac{1}{a}} \cdot (2a)^{\frac{1}{2a}} \cdot (4a)^{\frac{1}{4a}} \cdot (8a)^{\frac{1}{8a}} \cdots \infty$ is $\frac{8}{27}$,then the value of $a$ is:

The $n^{th}$ term of the series $1 + \frac{4}{5} + \frac{7}{5^2} + \frac{10}{5^3} + \dots$ will be