All-seeing eye

If the radius shrinks by the factor 1/2 for each smaller circle, what fraction of the whole area is represented by all the blue crescents (separated by the red crescents)? Assume the circles keep going inward forever.

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The blue fraction is 4/5.

Solution by Jason Elliot Benda

The area is proportional to the square of the linear measure, so it is (1/2)(1/2) each time, or 1/4. 1 – 1/4 + 1/16 … this is an infinite geometric series with first term 1 and common ratio –1/4, so the sum is (1) / (1 – (–1/4)) = 1 / (5/4) = 4/5.

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