Arbitrary Power of Complex Number/Lemma

Lemma for Arbitrary Power of Complex Number
Let $z = a + i b$ be a complex number.

Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Let $z^n = u_n + i v_n$.

Then $z^{n + 1} = u_{n + 1} + i v_{n + 1}$ where: