(a) State the sum of the series \(z+z^{2}+z^{3}+\ldots+z^{n}\), for \(z \neq 1\).

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