Sum of three complex numbers

Question

We have three complex numbers $z_1, z_2, z_3$ such that $|z_1|=|z_3|=|z_3|=1$ and $z_1+z_2+z_3=1$. Find $$S={z_1}^{2013}+{z_2}^{2013}+{z_3}^{2013}$$ I would like only a hint. I probably know everything elementary about complex numbers so just give me a little kick start

I thought about it. But I don't see how Moivre helps you here really... — asd11, Apr 09 '18 at 18:27

score 1 · Accepted Answer · answered Apr 09 '18 at 18:36

1

In order that both the conditions $|z_1|=|z_3|=|z_3|=1$ and $z_1+z_2+z_3=1$ are satisfied, we require that $z_1,z_2,z_3$ and $1$ form a quadrilateral of four equal sides, therefore a rhombus.

Without loss of generality, we can have $z_1=e^{i\theta}, z_2=1,z_3=-e^{i\theta}$

Can you finish this?

answered Apr 09 '18 at 18:36

David Quinn

34,121

That is nice. Thank you. I will take it from here! – asd11 Apr 09 '18 at 18:37

Sum of three complex numbers

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