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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 $S_n$?

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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.

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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:

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

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  • $\begingroup$ That assumes you know what $S_{n+1}$ is, which was not stated in the question. $\endgroup$ – nbubis Oct 29 '14 at 10:30
  • $\begingroup$ This is right. If you don't have $S_{n+1}$, you are stuck again. $\endgroup$ – Neraste Oct 29 '14 at 10:45

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