Let $(X_n : n \in \Bbb N)$ be a sequence of identically distributed random variables, with mean $ \mu$ and variance $\sigma^2 < \infty$. Set $S_0 = 0$ and $S_n=X_1+X_2+...+X_n$ for $n>0$. Let $N$ be a bounded, non-negative integer-valued random variable that is independent of the sequence $(X_n)$.
I've been asked to show that $\Bbb E(S_N^2 | N=n)=n\sigma^2+n^2\mu^2$, yet whenever I try to do this, I keep getting this conditional expectation to be $n^2\sigma^2+n^2\mu^2$. I've tried working from the definition of conditional expectation to no avail, as well as (what seems not very proper) considering simply $\Bbb E(S_n^2)$, but none seem to lead in the right direction.
All my methods seem to lead to $\sigma^2+\mu^2$ multiplied by some function of $N$ with little to do with the distributions of $X$. Any help with this would be much appreciated.