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Every rational appears on the sequence exactly once

Source: Iberoamerican Olympiad 2009, problem 5

September 24, 2009
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Problem Statement

Consider the sequence {an}n≥1 \{a_n\}_{n\geq1} defined as follows: a_1 \equal{} 1, a_{2k} \equal{} 1 \plus{} a_k and a_{2k \plus{} 1} \equal{} \frac {1}{a_{2k}} for every k≥1 k\geq 1. Prove that every positive rational number appears on the sequence {an} \{a_n\} exactly once.
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