Mathematics

A geometric sequence is completely determined by its first term $a$ and common ratio $r$—you have two unknowns and two pieces of information. The key insight is that the finite sum $S_5$ and infinite sum $S_\infty$ are related through the ratio $r$: the infinite sum "captures" what the tail of the series contributes beyond the first five terms. Set up equations using the formulas $S_5 = a\frac{1-r^5}{1-r}$ and $S_\infty = \frac{a}{1-r}$ (valid when $|r| < 1$), then solve the system to find $a$ and $r$.

A geometric sequence is completely determined by its first term $a$ and common ratio $r$—you have two unknowns and two pieces of information. The key insight is that the finite sum $S_5$ and infinite sum $S_\infty$ are related through the ratio $r$: the infinite sum "captures" what the tail of the series contributes beyond the first five terms. Set up equations using the formulas $S_5 = a\frac{1-r^5}{1-r}$ and $S_\infty = \frac{a}{1-r}$ (valid when $|r| < 1$), then solve the system to find $a$ and $r$.