Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.9 AIME Find the answer

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?

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

The area of the first square is 1, the area of the second is $\frac{1}{2}$ , the area of the third is $\frac{1}{4}$ , etc., so the answer is $1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots=2$ .

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