Commutator with powers of an eigenoperator ($[A,B]=cB\Rightarrow [A,B^n]=nc{B}^n$)

Suppose linear operators $A,B$ obey $[A,B]=cB$, where $c$ is a scalar, on a common invariant operator domain. The commutator product rule givesFor $n=0$ both sides vanish. This identity explains how powers of a creation operator shift the number operator eigenvalue.