$1-r$ unit in ring with $r^n = 0$

Question

Let $R$ be a ring with $r \in R$ and $r^n = 0$ for $n \in \mathbb{N}$. Show that $1-r$ is a unit in $R$.

I tried to use the geometric sum but I dont know how to proceed.

Answer 1

We have $(1-r)(1+r+\ldots+r^{n-1}) = 1$ if $r^n=0$.