Question 25
A sequence $(a_1, b_1),$ $(a_2, b_2),$ $(a_3, b_3),$ $\ldots$ of points in the coordinate plane satisfies $$
\begin{aligned} &(a_{n+1}, b_{n+1}) \\ &= \left(\sqrt{3}\,a_n - b_n,\; \sqrt{3}\,b_n + a_n\right) \\ &\quad (n = 1, 2, 3, \ldots) \end{aligned}
$$ Suppose that $(a_{100}, b_{100}) = (2, 4).$ What is $a_1 + b_1?$
\begin{aligned} &(a_{n+1}, b_{n+1}) \\ &= \left(\sqrt{3}\,a_n - b_n,\; \sqrt{3}\,b_n + a_n\right) \\ &\quad (n = 1, 2, 3, \ldots) \end{aligned}
$$ Suppose that $(a_{100}, b_{100}) = (2, 4).$ What is $a_1 + b_1?$