1. The problem is to find the common ratio $r$ in a geometric series given the sum formula $S_n = \frac{a_1}{1-r}$. 2. The formula $S_n = \frac{a_1}{1-r}$ is used for the sum of an infinite geometric series where $|r| < 1$. 3. To find $r$, you need to know the first term $a_1$ and the sum $S_n$. 4. Rearrange the formula to solve for $r$: $$S_n = \frac{a_1}{1-r} \implies 1-r = \frac{a_1}{S_n} \implies r = 1 - \frac{a_1}{S_n}$$ 5. This means you subtract the fraction $\frac{a_1}{S_n}$ from 1 to get $r$. 6. Make sure $S_n$ and $a_1$ are known values before calculating $r$. 7. Example: If $a_1 = 2$ and $S_n = 10$, then $$r = 1 - \frac{2}{10} = 1 - 0.2 = 0.8$$ This is how you find the common ratio $r$ from the sum formula.