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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