summation - Control ratio of geometric series through its sum

A geometric series $S_n$ is the sum of the $n$ first elements of a geometric sequence $u_n$:

$$u_n = ar^n \space \forall n \in \mathbb{N}^*$$

with $u_0$ defined, and:

$$S_n = \sum_{k = 0}^{k = n - 1}u_k=a\frac{1 - r^n}{1 - r}$$

Then, is there a way to determine the ratio $r$ analytically through a given finite $n$ and finite sum $S_n$?