150 Omg!

Algebra Level 5

\[ a_1 ^n + a_2 ^n + a_3 ^n + a_4 ^n + a_5 ^n + a_6 ^n + a_7 ^n \]

For positive integer \(n\), denote \(S_n\) as the value of the above expression, where \(a_1, a_2, a_3, a_4, a_5, a_6, a_7 \) are complex numbers.

Given that \(S_1 = S_2 = S_3 = S_4 = S_5 = 0 \), \(S_6 = 12 \), and \( \displaystyle \sum_{\text{cyclic}} a_1 a_2 a_3 a_4 a_5 a_6 = -2 \)

Find the value of \(x\) if \(S_{150} = 12x \) .

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