Help with problem: Let $w$ a 15th-primitive-root of unity. Find all $n \in \Bbb N_{<0}$ such that $\sum_{i=0}^{n-1} w^{5i}=0$

Question

we are starting to see complex numbers in my algebra class. So I have the following problem:

Let $w$ a 15th-primitive-root of unity. Find all $n \in \Bbb N_{<0}$ such that $\sum_{i=0}^{n-1} w^{5i}=0$

I thougth about seeing it as a geometric sum:

$\sum_{i=0}^{n-1} w^{5i}= \frac {(w^5)^n - 1}{w^5-1}$

But I don't know what should come next. As I'm just getting started with this subject I'm missing which properties from primitive roots should I use. Any suggestions?

Answer 1

Well, after you carried on the geometric sum, things become way simpler:

$$\sum_{i=0}^{n-1} w^{5i}=0\iff w^{5n}=1\stackrel{\text{Because primitive}}\iff 5n=15k\;,\;\;\text{for some}\;\;k\in\Bbb Z\iff$$

$$\iff n=3k\;,\;\;k\in\Bbb Z$$

Now choose the element as to belong to $\;\Bbb N_{<0}\;$ , which I presume means the negative integers, and that's all.