Sarika、Dev 和 Rajiv 正在分享一大块奶酪。他们轮流切下奶酪剩余部分的一半并吃掉：首先 Sarika 吃掉一半奶酪，接着 Dev 吃掉剩余部分的一半，随后 Rajiv 吃掉剩余部分的一半，然后又轮到 Sarika，如此循环往复，直到奶酪小到看不见为止。请问 Sarika 一共大约吃掉了这块奶酪的几分之几？

A. $\frac{4}{7}$

B. $\frac{3}{5}$

C. $\frac{2}{3}$

D. $\frac{3}{4}$

E. $\frac{7}{8}$

1. 2025 AMC 8 官方解析

Answer (A): In each round, Sarika eats twice as much cheese as Dev, and Dev eats twice as much cheese as Rajiv. Once they have eaten all the cheese, Dev will have eaten about twice as much cheese as Rajiv, and Sarika will have eaten about 4 times as much cheese as Rajiv. If $r$ is Rajiv's portion of cheese, then Sarika's portion is about $4r$ and Dev's portion is about $2r$. Together, the three will eat a portion approximately of size $7r$, so the fraction of the original block Sarika eats is about $\frac{4r}{7r} = \frac{4}{7}$ of the total amount.

OR

The following infinite geometric series represent each person's share of the cheese:

Sarika: $\frac{1}{2} + \frac{1}{16} + \frac{1}{128} + \cdots$

Dev: $\frac{1}{4} + \frac{1}{32} + \frac{1}{256} + \cdots$

Rajiv: $\frac{1}{8} + \frac{1}{64} + \frac{1}{512} + \cdots$

Using the infinite geometric series sum formula $S = \frac{a}{1 - r}$ with $a$ equal to the first term $\frac{1}{2}$ and ratio $r$ equal to $\frac{1}{8}$, the fraction of the block of cheese eaten by Sarika is about
$\frac{\frac{1}{2}}{1 - \frac{1}{8}} = \frac{\frac{1}{2}}{\frac{7}{8}} = \frac{1}{2} \cdot \frac{8}{7} = \frac{4}{7}.$