2022 ACJC P2 Q4

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]

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]

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]