Question

If z 2 + z + 1 = 0,
$z ∈ C$, then $\left| \sum_{n=1}^{15} \left( z_n + (-1)^n \frac{1}{z_n} \right)^2 \right|$
is equal to ________.

Answer 1

The correct answer is 2
∵ z 2 + z + 1 = 0
⇒ ω or ω2
∵$$\left| \sum_{n=1}^{15} \left( z_n + (-1)^n \frac{1}{z_n} \right)^2 \right|
=$\left| \sum_{n=1}^{15} z^{2n} + \sum_{n=1}^{15} z^{-2n} + 2.\sum_{n=1}^{15} (-1)^n \right|$
= |0 + 0 – 2|
= 2