Let $f$ be a function with the following properties:
(i) $f(1) = 1$, and
(ii) $f(2n) = n \cdot f(n)$ for any positive integer $n$.
What is the value of $f(2^{100})$?
$\textbf {(A)}\ 1 \qquad \textbf {(B)}\ 2^{99} \qquad \textbf {(C)}\ 2^{100} \qquad \textbf {(D)}\ 2^{4950} \qquad \textbf {(E)}\ 2^{9999}$
From (ii), note that
\begin{alignat*}{8} f(2) &= 1\cdot f(1) &&= 1, \\  f\left(2^2\right) &= 2\cdot f(2)  &&= 2, \\  f\left(2^3\right) &= 2^2\cdot f\left(2^2\right) &&= 2^{2+1}, \\ f\left(2^4\right) &= 2^3\cdot f\left(2^3\right) &&= 2^{3+2+1}, \end{alignat*}
and so on.
In general, we have \[f\left(2^n\right)=2^{(n-1)+(n-2)+(n-3)+\cdots+3+2+1}\] for any positive integer $n.$
Therefore, the answer is
\begin{align*} f\left(2^{100}\right)&=2^{99+98+97+\cdots+3+2+1} \\ &=2^{99\cdot100/2} \\ &= \boxed{\textbf {(D)}\ 2^{4950}}. \end{align*}
~MRENTHUSIASM