2008 AMC 12A Problem 25

A sequence (a_1,b_1), (a_2,b_2), (a_3,b_3), \ldots of points in the coordinate plane satisfies

(a_{n + 1}, b_{n + 1}) = (\sqrt {3}a_n - b_n, \sqrt {3}b_n + a_n) for n = 1,2,3,\ldots.

Suppose that (a_{100},b_{100}) = (2,4). What is a_1 + b_1?

\mathrm{(A)}\ -\frac{1}{2^{97}}\qquad\mathrm{(B)}\ -\frac{1}{2^{99}}\qquad\mathrm{(C)}\ 0\qquad\mathrm{(D)}\ \frac{1}{2^{98}}\qquad\mathrm{(E)}\ \frac{1}{2^{96}}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.