HMMT 二月 2001 · 团队赛 · 第 7 题
HMMT February 2001 — Team Round — Problem 7
题目详情
- The Fibonacci numbers are defined by F = F = 1 and F = F + F for n ≥ 1. 1 2 n +2 n +1 n The Lucas numbers are defined by L = 1, L = 2, and L = L + L for n ≥ 1. 1 2 n +2 n +1 n 15 ∏ F 2 n Fn n =1 Calculate . 13 ∏ L n n =1 sin 10+sin 20+sin 30+sin 40+sin 50+sin 60+sin 70+sin 80
解析
- The Fibonacci numbers are defined by F = F = 1 and F = F + F for n ≥ 1. 1 2 n +2 n +1 n The Lucas numbers are defined by L = 1, L = 2, and L = L + L for n ≥ 1. 1 2 n +2 n +1 n 15 ∏ F 2 n Fn n =1 Calculate . 13 ∏ L n n =1 F 2 n Solution: It is easy to show that L = , so the product above is L 4 L 5 = 843 · n 1 1 F n 1364 = 1149852 . sin 10+sin 20+sin 30+sin 40+sin 50+sin 60+sin 70+sin 80