2022 AMC 10A 第 1 题

Question

求下式的值：

3+\dfrac{1}{3+\dfrac{1}{3+\dfrac{1}{3}}}\text{？}

What is the value of

3+\dfrac{1}{3+\dfrac{1}{3+\dfrac{1}{3}}}?

\dfrac{31}{10}

\dfrac{49}{15}

\dfrac{33}{10}

\dfrac{109}{33}

\dfrac{15}{4}

从最里面的分母开始

Start from the innermost denominator

每次化简连分数的一层

Simplify one layer of the continued fraction at a time

这个连分数可以逐层化简如下：

\begin{aligned} 3 +& \dfrac{1}{3 + \dfrac{1}{3 + \dfrac{1}{3}}} \\ &= 3 + \dfrac{1}{3 + \dfrac{1}{\dfrac{10}{3}}} \\ &= 3 + \dfrac{1}{3 + \dfrac{3}{10}} \\ &= 3 + \dfrac{1}{\dfrac{33}{10}} \\ &= 3 + \dfrac{10}{33} \\ &= \dfrac{109}{33} \end{aligned}\text{。}

所以正确答案是 D。

We can simplify this expression as follows:

\begin{aligned} 3 +& \dfrac{1}{3 + \dfrac{1}{3 + \dfrac{1}{3}}} \\ &= 3 + \dfrac{1}{3 + \dfrac{1}{\dfrac{10}{3}}} \\ &= 3 + \dfrac{1}{3 + \dfrac{3}{10}} \\ &= 3 + \dfrac{1}{\dfrac{33}{10}} \\ &= 3 + \dfrac{10}{33} \\ &= \dfrac{109}{33}. \end{aligned}

Thus, D is the correct answer.

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