Hello, can I get some help with the following?
Show that if w is an nth root of unity (w != 1 and n > 1) then
w + w^2 + ...+ w^n = 0
Thanks!
If
Then
.
This means by DeMoivre's Theorem, the focus root is
.
But there must be exactly roots, all spaced evenly about a circle.
So that means any can be written as
, where is a positive integer no greater than .
That means
But any integer multiple of gives the same result as .
So that means .
Now let's go back to the original problem...
We have .
This is a geometric series with and .
So the sum is
.
But .
So that means the sum is
.
Q.E.D.