extrapolate terms of sequence, knowing sums

Question

I'm not familiar with filters etc, so I'm asking here if anyone knows an existing method to do the following. I have a sequence of observations $S_n$ and I know that they satisfy $S_n=\sum_{i=1}^m a(n,i)$ where $m$ is fixed an known, while $a(n,i)$ are unknown scalars. However I know that $$\frac{a(n+1,i)}{a(n,i)}\approx \frac{a(n,i)}{a(n-1,i)}$$ Knowing the sequence $S_n$ and the above relation, does anybody know a method for estimating the scalars $a(n,i)$? Thanks