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 $S_n$?


2 Answers 2


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.


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:

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

  • $\begingroup$ That assumes you know what $S_{n+1}$ is, which was not stated in the question. $\endgroup$
    – nbubis
    Oct 29, 2014 at 10:30
  • $\begingroup$ This is right. If you don't have $S_{n+1}$, you are stuck again. $\endgroup$
    – Neraste
    Oct 29, 2014 at 10:45

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.