lacunary - Mathnotes

First, define $S_N$ as

$$S_N=\sum _{k=0}^{N-1}{w}^k=1+w+w^2+\cdots +w^{N-1}.$$

Then

Subtracting the second equation from the first gives

because everything in the middle cancels out.

Now, we can factor out $1-w$ from the left hand side to get