# Fibonacci Complex

Algebra Level 5

Let $$a_0 = 0$$, $$a_1 = 1$$, and for $$n \geq 1$$, $a_{n+1} = -a_{n-1} + i\: a_n.$

Let $$\displaystyle S_n = \sum_{k=1}^{4n} a_k$$. Determine $$\displaystyle \lim_{n\to\infty} \text{arg}\ S_n$$.

(Give your answer in radians, between $$-\pi$$ and $$+\pi$$.)

Clarification: $$i = \sqrt{-1}$$.

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