Suppose that $X\sim \mathrm{Bin}(n,p)$, where $n = 10000$ and $p = 0.5$
What is $E(3^X)$?
I know that $E(3^X) = 2^{10000}$, but I have no clue how to prove this. And in fact, through trial and error, it seems that $E(3^X) = 2^n$.
How would one reduce this thing into $2^{10000}$:
$$E(3^x) = \sum_{i=0}^{10000} 3^i(0.5)^{10000}{10000 \choose i}$$
\begin{align} \sum_{i=0}^n3^i(0.5)^{1000}\binom{n}{i}&=(0.5)^n\sum_{i=0}^n3^i\binom{n}{i}\\ &=(0.5)^n\sum_{i=0}^n3^i\cdot 1^{n-i}\cdot \binom{n}{i}\\ &=(0.5)^n (3+1)^n\\ &=2^n \end{align}
where I have used $(a+b)^n = \sum_{i=0}^na^ib^{n-i}\binom{n}{i}$