Mathematics

The key insight is that in a geometric series, the sum formula $S = \frac{a}{1-r}$ directly connects the first term $a$, common ratio $r$, and sum. Since you're told $a = 2r$, you can substitute this relationship into the sum formula and solve for $r$ using the given convergence value. This turns a two-variable problem into one equation with one unknown.

The key insight is that in a geometric series, the sum formula $S = \frac{a}{1-r}$ directly connects the first term $a$, common ratio $r$, and sum. Since you're told $a = 2r$, you can substitute this relationship into the sum formula and solve for $r$ using the given convergence value. This turns a two-variable problem into one equation with one unknown.