Let $\omega = e^{2\pi i/6}$ (primitive 6th root of unity). Find $\sum_{k=1}^5 |1 - \omega^k|^2$.
Step-by-Step Solution
Key Concept: $|1 - \omega^k|^2 = 2 - 2 \cos(2\pi k/6)$. Sum from $k = 1$ to $5$: $\sum_{k=1}^5 (2 - 2 \cos(2\pi k/6)) = 10 - 2 \operatorname{Re}(\sum_{k=1}^5 \omega^k)$. Use the fact that $\sum_{k=0}^5 \omega^k = 0$.
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Correct Answer: 12