AMC12 2009 A

AMC12 2009 A · Q2

AMC12 2009 A · Q2. It mainly tests Fractions.

Which of the following is equal to $1 + \frac {1}{1 + \frac {1}{1 + 1}}$?

下列哪个等于 $1 + \frac {1}{1 + \frac {1}{1 + 1}}$？

(A) $\frac{5}{4}$ $\frac{5}{4}$

(B) $\frac{3}{2}$ $\frac{3}{2}$

(D) 2 2

(E) 3 3

Answer

Correct choice: (C)

正确答案：（C）

Solution

We compute: \begin{align*} 1 + \frac {1}{1 + \frac {1}{1 + 1}} &= 1 + \frac {1}{1 + \frac {1}{1 + 1}} \\ &= 1 + \frac {1}{1 + \frac 12} \\ &= 1 + \frac {1}{\frac 32} \\ &= 1 + \frac 23 \\ &= \frac 53 \end{align*} This is choice $\boxed{\text{C}}$. Interesting sidenote: The continued fraction $1 + \frac {1}{1 + \frac {1}{1 + 1....}}$ is equal to the golden ratio, or $\frac{1+\sqrt{5}}{2}$.

我们计算： \begin{align*} 1 + \frac {1}{1 + \frac {1}{1 + 1}} &= 1 + \frac {1}{1 + \frac {1}{1 + 1}} \\ &= 1 + \frac {1}{1 + \frac 12} \\ &= 1 + \frac {1}{\frac 32} \\ &= 1 + \frac 23 \\ &= \frac 53 \end{align*} 因此对应选项为 $\boxed{\text{C}}$。有趣的补充：连分数 $1 + \frac {1}{1 + \frac {1}{1 + 1....}}$ 等于黄金比例，即 $\frac{1+\sqrt{5}}{2}$。

Topics

AR.001 Fractions