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

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

If $S = \frac{7}{5} + \frac{9}{5^{2}} + \frac{13}{5^{3}} + \frac{19}{5^{4}} + \ldots$, then $160 \,S$ is equal to....... .

Let $p(x)=a_0+a_1 x+\ldots+a_n x^n$. If $p(-2)=-15, p(-1)=1, p(0)=7, p(1)=9, p(2)=13$ and $p(3)=25$,then the smallest possible value of $n$ is

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 $...........$.

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