For each positive integer $n$, the mean of the first $n$ terms of a sequence is $n$. What is the $2008$th term of the sequence?
$\textbf{(A)}\ 2008 \qquad \textbf{(B)}\ 4015 \qquad \textbf{(C)}\ 4016 \qquad \textbf{(D)}\ 4030056 \qquad \textbf{(E)}\ 4032064$
Letting $S_n$ be the nth partial sum of the sequence:
$\frac{S_n}{n} = n$
$S_n = n^2$
The only possible sequence with this result is the sequence of odd integers.
$a_n = 2n - 1$
$a_{2008} = 2(2008) - 1 = 4015 \Rightarrow \textbf{(B)}$