# 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 $$S_{150} = 12x$$ .

×