Approximate Solution to Backwards Recurrence of Dynamic Game

Question

Suppose we keep tossing a fair dice until we reach some cumulative sum greater than or equal to $N$. Then, let $S_k$ be the expected value of the final sum, given that the current sum is $k$. We have that $S_{N+j}=N+j$ for $0 \leq j \leq 5$. For $k<N$, we also have the recurrence \begin{equation*} S_k=\frac{1}{6} \sum_{j=1}^6 S_{k+j}. \end{equation*} This problem falls into the broader class of problems with a recurrence that has a finite but high base case. At this point, it is possible to write a program to compute $S_0$, the expected sum of the game, but I was wondering if there was any theory that gives good approximations/bounds for such recursions. The steady-state solution is $0$, and a characteristic polynomial approach is ugly.