Showing that $|z^{n+1}-z^n|=|z-1|$

Question

I need to show that $|z^{n+1}-z^n|=|z-1|$ where $z$ is a complex number, and the series $\{ a_{n} \} ^\infty _{n=1}$ is defined as $a_n=z^n$. It is assumed that $|z|=1$.

I'm having trouble showing that :/

Answer 1

Note that $$|z^{n+1} - z^n| = |z^n(z - 1)| = |z^n| \cdot |z - 1| = |z|^n \cdot |z - 1| = |z - 1|.$$