Question

萨里卡、德夫和拉吉夫分享一大块奶酪。他们轮流切下剩余奶酪的一半并吃掉：先萨里卡吃掉一半，然后德夫吃掉剩余奶酪的一半，然后拉吉夫吃掉此时剩余奶酪的一半，接着又轮到萨里卡，以此类推。他们一直吃到奶酪小到看不见为止。萨里卡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

小提示：

萨里卡吃的是连续二分序列中的每第三项

Sarika eats every third term of the halving sequence

大提示：

她吃到的份额构成公比为 $\frac{1}{8}$ 的等比级数

Her shares form a geometric series with ratio $\frac{1}{8}$

视频讲解：

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文字解答：

萨里卡第一步吃掉原奶酪的 $\frac{1}{2}$ 。

然后德夫吃掉原奶酪的 $\frac{1}{2^2}$ 。

接着拉吉夫吃掉原奶酪的 $\frac{1}{2^3}$ 。

萨里卡接下来又吃掉原奶酪的 $\frac{1}{2^4}$ 。

这个模式继续下去。最终萨里卡吃到原奶酪的比例为 $\frac{1}{2} + \frac{1}{2^4} + \frac{1}{2^7} + \dots$ 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 $\frac{a}{1-r} = \frac{\frac{1}{2}}{1 - \frac{1}{8}} = \frac{\frac{1}{2}}{\frac{7}{8}} = \frac{4}{7}$ 所以选 A。

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