- #1

DEMJ

- 44

- 0

## Homework Statement

If c is any nth root of unity other than 1, then

[tex] 1 + c + c^2 + \cdots + c^{n-1} = 0[/tex]

## The Attempt at a Solution

This is what is done so far and I am at a dead stall for about 2 hours lol. Any ideas on what I should be thinking of next? Should I continue simplifying? I have tried to continue simplifying but it always leads to nothing relevant. Do I need to be using [tex]re^{i\theta}[/tex] ?

Proof:

Assume [tex]c^n = 1[/tex] and [tex]c \not = 1[/tex]. Then [tex]c^n -1 = 0[/tex]. Note that

[tex](c-1)(1 + c + c^2 + \cdots + c^{n-1}) = (c)(1 + c + c^2 + \cdots c^{n-1}) + (-1)(1 + c + c^2 + \cdots + c^{n-1}) = c + c^2 + c^3 + \cdots + c^n - 1 - c - c^2 - \cdots - c^{n-1}[/tex]