HMMT 二月 2002 · 冲刺赛 · 第 8 题
HMMT February 2002 — Guts Round — Problem 8
题目详情
英文原题
- [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 themidpoints of the sides of the third square to form a fourth square. And so forth. What isthe sum of the areas of all the squares in this infinite series?
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解析
英文解析
- 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 themidpoints of the sides of the third square to form a fourth square. And so forth. What isthe sum of the areas of all the squares in this infinite series?
Solution: The area of the first square is 1, the area of the second is , the area of the 1
1 1 1 12
third is , etc., so the answer is 1 + + + + · · · = 2 .
4 2 4 8
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