HMMT 二月 2001 · CALC 赛 · 第 1 题
HMMT February 2001 — CALC Round — Problem 1
题目详情
- A sequence of ants walk from (0 , 0) to (1 , 0) in the plane. The n th ant walks along 1 n semicircles of radius with diameters lying along the line from (0 , 0) to (1 , 0). Let L be n n the length of the path walked by the n th ant. Compute lim L . n n →∞ 5 3
解析
- A sequence of ants walk from (0 , 0) to (1 , 0) in the plane. The n th ant walks along 1 n semicircles of radius with diameters lying along the line from (0 , 0) to (1 , 0). Let L be n n the length of the path walked by the n th ant. Compute lim L . n n →∞ ( ) 1 1 2 π Solution: A semicircle of radius has length π = , so n such semicircles have n 2 n n total length π . 5 3