2018 AMC 8 第 2 题

先试着解答 2018 AMC 8 第 2 题,然后核对你的答案与视频与精心整理的解答,解答来自 LIVE by Po-Shen Loh。你也可以参加完整限时模拟考试、查看全部 2018 AMC 8 解答,或核对答案。

所有题目均经美国数学协会(MAA)官方合法授权使用。

2.

下列乘积的值是多少?(1+11)⋅(1+12)⋅(1+13)⋅(1+14)⋅(1+15)⋅(1+16)?\begin{align*} &\left(1+\frac{1}{1}\right)\cdot\left(1+\frac{1}{2}\right)\\ &\quad{}\cdot\left(1+\frac{1}{3}\right)\cdot\left(1+\frac{1}{4}\right)\\ &\quad{}\cdot\left(1+\frac{1}{5}\right)\cdot\left(1+\frac{1}{6}\right) \end{align*}\text{?}
What is the value of the product (1+11)⋅(1+12)⋅(1+13)⋅(1+14)⋅(1+15)⋅(1+16)?\begin{align*} &\left(1+\frac{1}{1}\right)\cdot\left(1+\frac{1}{2}\right)\\ &\quad{}\cdot\left(1+\frac{1}{3}\right)\cdot\left(1+\frac{1}{4}\right)\\ &\quad{}\cdot\left(1+\frac{1}{5}\right)\cdot\left(1+\frac{1}{6}\right)? \end{align*}
76\dfrac{7}{6}
43\dfrac{4}{3}
72\dfrac{7}{2}
77
88
答案:D
知识点:裂项相消分数
小提示:
把每个因子 1+1n1+\frac1n 改写成 n+1n\frac{n+1}{n}
Rewrite each factor 1+1n1+\frac1n as n+1n\frac{n+1}{n}.
大提示:
改写后相邻分子和分母会相消
The product telescopes after the fractions are rewritten.
视频讲解:
解答视频缩略图
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文字解答:
先注意,形如 1+1n1 + \frac{1}{n} 的因子可以改写为 nn+1n=n+1n。\frac{n}{n} + \frac{1}{n} = \frac{n+1}{n}\text{。}因此题中的乘积可以写成 (1+11)⋅(1+12)⋅(1+13)⋅(1+14)⋅(1+15)⋅(1+16)=21⋅32⋅43⋅54⋅65⋅76=7\begin{align*} &\left(1+\frac{1}{1}\right)\cdot\left(1+\frac{1}{2}\right)\\ &\quad{}\cdot\left(1+\frac{1}{3}\right)\cdot\left(1+\frac{1}{4}\right)\\ &\quad{}\cdot\left(1+\frac{1}{5}\right)\cdot\left(1+\frac{1}{6}\right)\\ &=\dfrac{2}{1} \cdot \dfrac{3}{2} \cdot \dfrac{4}{3} \cdot\dfrac{5}{4} \cdot\dfrac{6}{5} \cdot\dfrac{7}{6}\\ &=7 \end{align*} 所以正确答案是 D。
Let’s first note that if we are given an expression of the form 1+1n,1 + \frac{1}{n}, we can rewrite this as nn+1n=n+1n.\frac{n}{n} + \frac{1}{n} = \frac{n+1}{n}. With that in mind, we can rewrite the expression given to us in the problem, as shown below: (1+11)⋅(1+12)⋅(1+13)⋅(1+14)⋅(1+15)⋅(1+16)=21⋅32⋅43⋅54⋅65⋅76=7\begin{align*} &\left(1+\frac{1}{1}\right)\cdot\left(1+\frac{1}{2}\right)\\ &\quad{}\cdot\left(1+\frac{1}{3}\right)\cdot\left(1+\frac{1}{4}\right)\\ &\quad{}\cdot\left(1+\frac{1}{5}\right)\cdot\left(1+\frac{1}{6}\right)\\ &=\dfrac{2}{1} \cdot \dfrac{3}{2} \cdot \dfrac{4}{3} \cdot\dfrac{5}{4} \cdot\dfrac{6}{5} \cdot\dfrac{7}{6}\\ &=7 \end{align*} Thus, the correct answer is D.
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