2002 Putnam A5

Define a sequence by $a_0=1$, together with the rules

$a_{2n+1} = a_n$ and $a_{2n+2} = a_n + a_{n+1}$ for each

integer $n \geq 0$. Prove that every positive rational number

appears in the set

$$

\left\{ \frac{a_{n-1}}{a_n}: n \geq 1 \right\} =

\left\{ \frac{1}{1}, \frac{1}{2}, \frac{2}{1}, \frac{1}{3},

\frac{3}{2}, \dots \right\}.

$$

通过$a_0=1$定义一个序列，并附上规则

$a_{2n+1} = a_n$ 和 $a_{2n+2} = a_n + a_{n+1}$

整数 $n \geq 0$。证明每个正有理数

出现在集合中

$$

\left\{ \frac{a_{n-1}}{a_n}: n \geq 1 \right\} =

\left\{ \frac{1}{1}, \frac{1}{2}, \frac{2}{1}, \frac{1}{3},

\frac{3}{2}、\点\right\}。

$$