Sum geometric series with complex terms

A question is this type if and only if it asks to find the sum of a finite or infinite geometric series involving complex exponentials or trigonometric functions, often separating real and imaginary parts.

26 questions · Challenging +1.3

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AQA Further Paper 2 2023 June Q15
10 marks Challenging +1.2
15
  1. Given that \(z = \cos \theta + \mathrm { i } \sin \theta\), use de Moivre's theorem to show that $$z ^ { n } - z ^ { - n } = 2 i \sin n \theta$$ 15
  2. The series \(S\) is defined as $$S = \sin \theta + \sin 3 \theta + \ldots + \sin ( 2 n - 1 ) \theta$$ Use part (a) to express \(S\) in the form $$S = \frac { 1 } { 2 \mathrm { i } } \left( G _ { 1 } \right) - \frac { 1 } { 2 \mathrm { i } } \left( G _ { 2 } \right)$$ where each of \(G _ { 1 }\) and \(G _ { 2 }\) is a geometric series.
    15
  		3. Hence, show that[5 marks]