HMMT 二月 2002 · 冲刺赛 · 第 8 题
HMMT February 2002 — Guts Round — Problem 8
题目详情
- [7] Draw a square of side length 1. Connect its sides’ midpoints to form a second square. Connect the midpoints of the sides of the second square to form a third square. Connect the midpoints of the sides of the third square to form a fourth square. And so forth. What is the sum of the areas of all the squares in this infinite series? √
解析
- Draw a square of side length 1. Connect its sides’ midpoints to form a second square. Connect the midpoints of the sides of the second square to form a third square. Connect the midpoints of the sides of the third square to form a fourth square. And so forth. What is the sum of the areas of all the squares in this infinite series? 1 Solution: The area of the first square is 1, the area of the second is , the area of the 2 1 1 1 1 third is , etc., so the answer is 1 + + + + · · · = 2 . 4 2 4 8 √