WAEC Further Mathematics 2020 Paper

Concept:The formula $S = \frac{a}{1 - r}$ gives the sum of an infinite geometric series only when the common ratio satisfies $|r| < 1$.Explanation:Find the common ratio $r$ for each series and check whether $|r| < 1$.For series A, $4 + 8 + 16 + 32 + \cdots$ has $r = \frac{8}{4} = 2$.Since $|2| > 1$, the formula cannot be used here.For series B, $\frac{1}{2} + \frac{5}{2} + \frac{25}{2} + \cdots$ has $r = 5$.Since $|5| > 1$, this series is not valid for the formula.For series C, $\frac{4}{81} + \frac{2}{27} + \frac{1}{9} + \cdots$ has $r = \frac{3}{2}$.Since $|\frac{3}{2}| > 1$, the formula does not apply here.For series D, $128 + 64 + 32 + 16 + \cdots$ has $r = \frac{64}{128} = \frac{1}{2}$.Since $|\frac{1}{2}| < 1$, the infinite series converges and the formula $S = \frac{a}{1 - r}$ can be used.Therefore, only series D satisfies the required condition.Answer:D. $128 + 64 + 32 + 16 + \cdots$