2008 AMC 10A 第 5 题

Question

下列哪一项等于乘积

\begin{aligned} &\dfrac{8}{4} \cdot \dfrac{12}{8} \cdot \dfrac{16}{12} \cdots \dfrac{4n+4}{4n} \\ &\quad \cdots \dfrac{2008}{2004} \end{aligned}\text{？}

Which of the following is equal to the product

\begin{aligned} &\dfrac{8}{4} \cdot \dfrac{12}{8} \cdot \dfrac{16}{12} \cdots \dfrac{4n+4}{4n} \\ &\quad \cdots \dfrac{2008}{2004}? \end{aligned}

251

502

1004

2008

4016

写出前几个因子，寻找约分规律。

Write out the first few factors and look for cancellation

每个分子都会与下一个分母约掉，最后剩下

\dfrac{2008}{4}

。

Each numerator cancels the next denominator, leaving

\dfrac{2008}{4}

除第一个分母外，每个分母都与前一个分数的分子约掉，因此整个乘积逐项约消为

\dfrac{2008}{4} = 502

。

所以正确答案是 B。

Every denominator except the first cancels with the numerator of the previous fraction, so the whole product telescopes to

\dfrac{2008}{4} = 502.

Thus, the correct answer is B.

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