If $w = e^{2i\pi/5} $, then $1 + w + w^{2} + w^{3} + 5w^{4} + 4w^{5} + 4w^{6} + 4w^{7} + 4w^{8} + 5w^{9}$=?

Question

If $w = e^{i\frac{2\pi}5} $, then $1 + w + w^{2} + w^{3} + 5w^{4} + 4w^{5} + 4w^{6} + 4w^{7} + 4w^{8} + 5w^{9}$ =?

I substituted $w$ into the expression and combined similar terms. I then tried to see which terns had real or imaginary parts that would cancel. That didn't work out. There's an annoying trick in this problem.

Answer 1

The trick is that $w^5 = 1$ and $w \neq 1$ implies that

$$1+w+w^2+w^3+w^4 = 0.$$

Use that twice.