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

Let $f(x) = \sin x + 2\sin^2 x + 3\sin^3 x + 4\sin^4 x + \dots \infty$. Then the number of solutions of the equation $f(x) = 2$ in $x \in [-\pi, \pi] - \{\pm \frac{\pi}{2}\}$ is

$1 + 2 \cdot 2 + 3 \cdot 2^2 + 4 \cdot 2^3 + \dots + 100 \cdot 2^{99} = \dots$

If $7 = 5 + \frac{1}{7}(5 + \alpha) + \frac{1}{7^2}(5 + 2\alpha) + \frac{1}{7^3}(5 + 3\alpha) + \dots \infty$,then the value of $\alpha$ is:

If the $r^{th}$ term of a series is $(2r + 1)2^{-r}$,what is the sum of its infinite terms?

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