HMMT 二月 2004 · 冲刺赛 · 第 19 题
HMMT February 2004 — Guts Round — Problem 19
题目详情
英文原题
- [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
英文解析
- 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 nn +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 itholds 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