Set $w=\cos\frac{\pi}{5}+i\sin\frac{\pi}{5}$.
I have to calculate:
$$1 + \sum_1^9 w^n$$
I have calculated that the answer is 0.
However, I am supposed to arrive at this conclusion without calculations and am not sure how to. Any ideas? Hints are appreciated over solutions.
Kind regards.
Note that $\omega$ given is a primitive $10$th root of unity. Then we can easily prove that $\omega^n, n \ne 10$ gives us all other primitive $10$th roots of unity. Then from vieta, we know that the sum over all $10$th roots of unity is $0$ and $1$ is the other $10$th root of unity. So the answer follows.