Mathematics

A geometric progression grows by multiplying by a fixed ratio each time, so the fifth term is just the first four terms' sum multiplied by that ratio. The key insight is that $S_5 = S_4 \cdot r$, where $r$ is the common ratio—this lets you find $r$ directly without needing to solve for individual terms first.

A geometric progression grows by multiplying by a fixed ratio each time, so the fifth term is just the first four terms' sum multiplied by that ratio. The key insight is that $S_5 = S_4 \cdot r$, where $r$ is the common ratio—this lets you find $r$ directly without needing to solve for individual terms first.