Finding the common ratio from adjacent elements in a geometric sum Is there a general solution for this problem?
Given $S_N$ and $S_{N+1}$ are the sums of geometric series, can we find the common ratio(s). (Assuming $a_0 = 1$).
To be more explicit, given
$$S_N = 1 + r + \ldots + r^{N-1} = \frac{1-r^N}{1-r}$$
and
$$S_{N+1} = 1 + r + \ldots + r^{N} = \frac{1-r^{N+1}}{1-r}\ ,$$
find $r$.
For example, given $S_N = 3$ and $S_{N+1} = 7$, find $r$.
In this example, $r=2$ is a solution, since for $r=2$ and $N=2$, $S_N = 1 + 2 = 3$ and $S_{N+1} = 1 + 2 + 4 = 7$.

We have
$$S_{N+1}-1=r+r^2+\cdots+r^N=rS_N\ .$$
So if $S_N$ and $S_{N+1}$ are given you can easily find
$$r=\frac{S_{N+1}-1}{S_N}\ .$$
