The sum of infinite terms of the following series $1 + \frac{4}{5} + \frac{7}{5^2} + \frac{10}{5^3} + \dots$ will be

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

If $u_{0}=8, u_{1}=3, u_{2}=12, u_{3}=51$,then the value of $\Delta^{3} u_{0}$ is

If $x, y, z$ are three positive numbers in $G.P.$,then $\frac{1 + \ln x}{2}, \frac{1 + \ln y}{4}, \frac{1 + \ln z}{8}$ are in

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

$1 + \frac{3}{2} + \frac{5}{2^2} + \frac{7}{2^3} + \dots \infty$ is equal to