If $(10)^9 + 2(11)^1(10)^8 + 3(11)^2(10)^7 + \dots + 10(11)^9 = k(10)^9$,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:

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

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

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

If $S(x) = (1+x) + 2(1+x)^2 + 3(1+x)^3 + \ldots + 60(1+x)^{60}$,$x \neq 0$,and $(60)^2 S(60) = a(b)^b + b$ where $a, b \in N$,then $(a+b)$ is equal to: