AMC 12 2008a 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 $a_1 + b_1$?

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