2018 AMC 10A 第 1 题

Question

求下式的值：

\left(\left((2+1)^{-1}+1\right)^{-1}+1\right)^{-1}+1\text{？}

What is the value of

\left(\left((2+1)^{-1}+1\right)^{-1}+1\right)^{-1}+1?

\dfrac{5}{8}

\dfrac{11}{7}

\dfrac{8}{5}

\dfrac{18}{11}

\dfrac{15}{8}

从最内层的倒数开始，由内向外计算。

Work from the innermost reciprocal outward

每取一次倒数，先化简，再加下一个

1

。

After each reciprocal step, simplify before adding the next

1

由内向外逐步化简：

\begin{aligned} &\left(\left((2+1)^{-1}+1\right)^{-1}+1\right)^{-1}+1 \\ &=\left(\left(\dfrac{1}{3}+1\right)^{-1}+1\right)^{-1}+1 \\ &=\left(\dfrac{3}{4}+1\right)^{-1}+1 \\ &=\dfrac{4}{7} + 1 \\ &=\dfrac{11}{7} \end{aligned}

因此正确答案是 B。

We can simplify this as follows.

\begin{aligned} &\left(\left((2+1)^{-1}+1\right)^{-1}+1\right)^{-1}+1 \\ &=\left(\left(\dfrac{1}{3}+1\right)^{-1}+1\right)^{-1}+1 \\ &=\left(\dfrac{3}{4}+1\right)^{-1}+1 \\ &=\dfrac{4}{7} + 1 \\ &=\dfrac{11}{7} \end{aligned}

Thus, B is the correct answer.

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