HMMT 二月 2004 · 冲刺赛 · 第 19 题

HMMT February 2004 — Guts Round — Problem 19

[8] The Fibonacci numbers are defined by F = F = 1, and F = F + F for 1 2 n n − 1 n − 2 n ≥ 3. If the number F F 2003 2004 − F F 2002 2003 is written as a fraction in lowest terms, what is the numerator?

解析

The Fibonacci numbers are defined by F = F = 1, and F = F + F for n ≥ 3. 1 2 n n − 1 n − 2 If the number F F 2003 2004 − F F 2002 2003 is written as a fraction in lowest terms, what is the numerator? Solution: 1 2 2 Before reducing, the numerator is F − F F . We claim F − F F = 2002 2004 n − 1 n +1 2003 n n +1 ( − 1) , which will immediately imply that the answer is 1 (no reducing required). This claim is straightforward to prove by induction on n : it holds for n = 2, and if it holds for some n , then 2 2 n +1 n +2 F − F F = F ( F + F ) − F ( F + F ) = F F − F = − ( − 1) = ( − 1) . n n +2 n +1 n − 1 n n n n +1 n +1 n − 1 n +1 n