2022 ACJC P2 Q4
(i)
It is given that ${{U}_{n}}=\cos \left[ \left( 2n+1 \right)\theta \right]$, for $n\ge 0$.
Show that for $n\ge 1$, ${{U}_{n}}+{{U}_{n-1}}=2\cos \left( 2n\theta \right)\cos \theta $.
[1]
(ii)
Hence show that $\sum\limits_{n=1}^{2N}{\left[ {{\left( -1 \right)}^{n+1}}\cos \left( 2n\theta \right) \right]=\frac{1}{2}\left( 1-\frac{\cos \left[ \left( 4N+1 \right)\theta \right]}{\cos \theta } \right)}$.
[3]
(iii)
Without the use of the graphic calculator, find the value of $\sum\limits_{n=11}^{41}{\left[ {{\left( -1 \right)}^{n+1}}\cos \frac{n\pi }{3} \right]}$, showing your working clearly.
[4]
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