For this question, please reference the image at:

All circles in the dartboard have the same center.
Each circle has half the radius of the previous circle.
The colors alternate orange-white-orange-white-etc. forever.

How much of the dartboard is colored orange?

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  1. the radii form a geometric sequence
    r, r/2, r/4, ...
    The orange rings have areas that form another sequence:
    πr^2 - π(r/2)^2, π(r/4)^2 - π(r/8)^2, ...
    = πr^2 (3/4 + 3/64 + ...)
    the sum of all those areas is thus
    πr^2 (3/4)/(1 - 1/16) = 4/5 πr^2
    so, 4/5 of the dartboard is orange

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