Control ratio of geometric series through its sum

Question

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 $S_n$?

Answer 1

That depends on $n$. Since $S_n$ is a polynomial of degree $n-1$, $$S_n=a(1+r+r^2+\ldots+r^{n-1})$$ By the Abel–Ruffini theorem, when $n>5$, there is no general solution that can be expressed by radicals.

Answer 2

Found the answer of my question, Rearrange the formula for the sum of a geometric series to find the value of its common ratio?, just in related:

$$a = S_{n+1} - rS_n$$