Question

下列乘积的值是多少？

$\begin{align*}\displaystyle & \left(\dfrac{1\cdot 3}{2\cdot 2}\right)\left(\dfrac{2\cdot 4}{3\cdot 3}\right)\left(\dfrac{3\cdot 5}{4\cdot 4}\right) \\ & \dots\left(\dfrac{97\cdot 99}{98\cdot 98}\right)\left(\dfrac{98\cdot 100}{99\cdot 99}\right) \end{align*}$

What is the value of the product below?

$\begin{align*}\displaystyle & \left(\dfrac{1\cdot 3}{2\cdot 2}\right)\left(\dfrac{2\cdot 4}{3\cdot 3}\right)\left(\dfrac{3\cdot 5}{4\cdot 4}\right) \\ & \dots\left(\dfrac{97\cdot 99}{98\cdot 98}\right)\left(\dfrac{98\cdot 100}{99\cdot 99}\right) \end{align*}$

$\displaystyle \dfrac{1}{2}$

$\displaystyle \dfrac{50}{99}$

$\displaystyle \dfrac{9800}{9801}$

$\displaystyle \dfrac{100}{99}$

$50$

答案：B

知识点：裂项相消分数

难度评级：1310

视频讲解：

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文字解答：

可以把所有因子重新组合为 $\begin{align*} & \dfrac{1}{2} \cdot \left(\dfrac{3 \cdot 2}{2\cdot 3}\right)\left(\dfrac{4\cdot 3}{3\cdot 4}\right) \\ & \dots\left(\dfrac{99\cdot 98}{98\cdot 99}\right) \cdot \dfrac{100}{99} \end{align*}$

由这个形式可见，中间各项全部相消，只剩 $\dfrac{1}{2} \cdot \dfrac{100}{99} = \dfrac{50}{99}.$

正确答案是 B。

We can regroup all the factors as follows. $\begin{align*} & \dfrac{1}{2} \cdot \left(\dfrac{3 \cdot 2}{2\cdot 3}\right)\left(\dfrac{4\cdot 3}{3\cdot 4}\right) \\ & \dots\left(\dfrac{99\cdot 98}{98\cdot 99}\right) \cdot \dfrac{100}{99} \end{align*}$

From this representation, we can see that all the middle terms cancel leaving only $\dfrac{1}{2} \cdot \dfrac{100}{99} = \dfrac{50}{99}.$

Thus, the correct answer is B.

2019 AMC 8 第 17 题

答案：B

难度评级：1310

视频讲解：

文字解答：

其他年份的第 17 题