Question 5 - A-Level Maths

State the sum of the series $z + z ^ { 2 } + z ^ { 3 } + \ldots + z ^ { n }$, for $z \neq 1$.
Given that $z$ is an $n$th root of unity and $z \neq 1$, deduce that $1 + z + z ^ { 2 } + \ldots + z ^ { n - 1 } = 0$.
Given instead that $z = \frac { 1 } { 3 } ( \cos \theta + \mathrm { i } \sin \theta )$, use de Moivre's theorem to show that $$\sum _ { m = 1 } ^ { \infty } 3 ^ { - m } \cos m \theta = \frac { 3 \cos \theta - 1 } { 10 - 6 \cos \theta }$$