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10:14 9231_w20_qp_22.pdf 7 (a) Show that \( \sum_{r=1}^{n} z^{2 r}=\frac{z^{2 n+1}-z}{z-z^{-1}} \), for \( z \neq 0,1,-1 \). \( [2] \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) C UCL.ES 2020 \( 9231 / 22 / 0 / \mathrm{N} / 20 \) 9 (b) By letting \( z=\cos \theta+i \sin \theta \), show that, if \( \sin \theta \neq 0 \), \[ 1+2 \sum_{r=1}^{n} \cos (2 r \theta)=\frac{\sin (2 n+1) \theta}{\sin \theta} \] \( \qquad \) 9231_w20_ms_22.pdf C UCLES 2020 Page 12 of 16

          10:14
9231_w20_qp_22.pdf
7 (a) Show that \( \sum_{r=1}^{n} z^{2 r}=\frac{z^{2 n+1}-z}{z-z^{-1}} \), for \( z \neq 0,1,-1 \).
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C UCL.ES 2020
\( 9231 / 22 / 0 / \mathrm{N} / 20 \)
9
(b) By letting \( z=\cos \theta+i \sin \theta \), show that, if \( \sin \theta \neq 0 \),
\[
1+2 \sum_{r=1}^{n} \cos (2 r \theta)=\frac{\sin (2 n+1) \theta}{\sin \theta}
\]
\( \qquad \)
9231_w20_ms_22.pdf
C UCLES 2020
Page 12 of 16

10:14
9231w20qp22.pdf
7 (a) Show that ∑r=1^n z^2 r=(z^2 n+1-z)/(z-z^-1), for z ≠ 0,1,-1.
[2]

C UCL.ES 2020
9231 / 22 / 0 / N / 20
9
(b) By letting z=cosθ+i sinθ, show that, if sinθ≠ 0,

1+2 ∑r=1^ncos (2 r θ)=(sin (2 n+1) θ)/(sinθ)

9231w20ms22.pdf
C UCLES 2020
Page 12 of 16

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