If $(20)^{19} + 2(21)(20)^{18} + 3(21)^2(20)^{17} + \ldots + 20(21)^{19} = k (20)^{19}$,then $k$ 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:

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

For a certain function $u_{x}$,given that $u_{0}=3, u_{1}=12, u_{2}=81, u_{3}=200, u_{4}=100, u_{5}=8$,then $\Delta^{5} u_{x}$ is equal to

The sum $\sum\limits_{k = 1}^{20} {k\frac{1}{{{2^k}}}} $ is equal to