Question

Sarika、Dev 和 Rajiv 分享一大块奶酪。他们轮流切下剩余奶酪的一半并吃掉：先 Sarika 吃掉一半，然后 Dev 吃掉剩下一半的一半，然后 Rajiv 吃掉剩下的一半，然后又轮到 Sarika，以此类推。他们一直吃到奶酪小到看不见为止。Sarika 总共大约吃掉了原奶酪的几分之几？

Sarika, Dev, and Rajiv are sharing a large block of cheese. They take turns cutting off half of what remains and eating it: first Sarika eats half of the cheese, then Dev eats half of the remaining half, then Rajiv eats half of what remains, then back to Sarika, and so on. They stop when the cheese is too small to see. About what fraction of the original block of cheese does Sarika eat in total?

$\dfrac{4}{7}$

$\dfrac{3}{5}$

$\dfrac{2}{3}$

$\dfrac{3}{4}$

$\dfrac{7}{8}$

答案：A

知识点：等比数列求和

难度评级：1580

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文字解答：

Sarika 第一步吃掉 $\frac{1}{2}$ 块奶酪。

然后 Dev 吃掉 $\frac{1}{2^2}$ 块奶酪。

接着 Rajiv 吃掉 $\frac{1}{2^3}$ 块奶酪。

Sarika 接下来又吃掉 $\frac{1}{2^4}$ 块奶酪。

这个模式继续下去。最终 Sarika 吃到的是这是首项 $a = \frac{1}{2}$ 、公比 $r = \frac{1}{2^3}$ 的无穷等比级数。其和为选 A。 $\frac{1}{2} + \frac{1}{2^4} + \frac{1}{2^7} + \dots$ $\frac{a}{1-r} = \frac{\frac{1}{2}}{1 - \frac{1}{8}} = \frac{\frac{1}{2}}{\frac{7}{8}} = \frac{4}{7}$

Sarika gets $\frac{1}{2}$ of the cheese in the first step.

Then Dev gets $\frac{1}{2^2}$ of the cheese.

Then Rajiv gets $\frac{1}{2^3}$ of the cheese.

Sarika then gets another $\frac{1}{2^4}$ of the cheese.

This pattern continues. Ultimately, Sarika gets $\frac{1}{2} + \frac{1}{2^4} + \frac{1}{2^7} + \dots$ which is the sum of an infinite geometric series with first term $a = \frac{1}{2}$ and common ratio $r = \frac{1}{2^3}.$ That sum is $\frac{a}{1-r} = \frac{\frac{1}{2}}{1 - \frac{1}{8}} = \frac{\frac{1}{2}}{\frac{7}{8}} = \frac{4}{7}$ which is choice A.

2025 AMC 8 第 20 题

答案：A

难度评级：1580

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