Question

For this question, please reference the image at:
imgur<dot>com/KSYAN6T



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?

Answer 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