Fractal Circle Area
A fractal pattern (pictured above) contains infinitely many circles, with each successive circle in the pattern having a radius 90% as big as the one that came before. If the area of the first (and largest) circle in the pattern is 1, what is the combined area of all of the circles?
Give your answer to 3 decimal places.
The answer is 5.263.
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1 solution
This is a geometric series whose sum is 1 / (1 - r). Because the area is proportional to the square of the radius, the common ratio is 0.81. 1 / (1 - 0.81) = 5.263.