What is the sum of the infinite series $1 + \frac{4}{5} + \frac{7}{5^2} + \frac{10}{5^3} + \dots$?

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

The sum of the series $1 + 2x + 3x^2 + 4x^3 + \dots$ up to $n$ terms is:

If $\tan \theta = \sqrt{\frac{3}{2}}$,the sum of the infinite series $1 + 2(1 - \cos \theta) + 3(1 - \cos \theta)^2 + 4(1 - \cos \theta)^3 + \dots \infty$ is

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 sum of $1 + \frac{2}{5} + \frac{3}{5^2} + \frac{4}{5^3} + \dots$ up to $n$ terms is