Show $\frac{k+2}{k(k+1)(k+2)}=\frac{1}{k(k+1)}=\frac{1}{k}-\frac{1}{k+1}$.

Hence show $X_n=1-\frac{1}{n+1}=\frac{n}{n+1}$.

Show the equivalent form $X_n=\frac{1}{2}+\frac{n-1}{2(n+1)}$.

Prove by induction that $X_n=\frac{n}{n+1}$ for all $n\in {\mathbb{Z}}^{+}$.

Evaluate $\sum _{k=12}^{50}\frac{k+2}{k(k+1)(k+2)}$.

Show $\frac{1}{(k+1)(k+3)}=\frac{1}{2}\left(\frac{1}{k+1}-\frac{1}{k+3}\right)$.

Hence obtain a closed form for $Y_n$.

Show the equivalent form $Y_n=\frac{n(n+5)}{4(n+2)(n+3)}+\frac{1}{6}$.

Prove by induction that $Y_n=\frac{5}{12}-\frac{2n+5}{2(n+2)(n+3)}$ holds for all $n\in {\mathbb{Z}}^{+}$.

Evaluate $\sum _{k=8}^{30}\frac{1}{(k+1)(k+3)}$ exactly.

Show that $\frac{2}{k(k+1)}=\frac{2}{k}-\frac{2}{k+1}$.

Hence show $W_n=\frac{2n}{n+1}$.

Show the equivalent form $W_n=1+\frac{n-1}{n+1}$.

Prove by induction that $W_n=\frac{2n}{n+1}$ for all $n\in {\mathbb{Z}}^{+}$.

Find $\sum _{k=15}^{40}\frac{2}{k(k+1)}$.

Prove that $z+\frac{1}{z}=2\cos \theta$, and hence that $z+\frac{1}{z}$ is real.

Using De Moivre’s theorem, show that for any $n\in \mathbb{N}$, $$z^n+\frac{1}{z^n}=2\cos (n\theta ).$$

Hence solve $$z^6+\frac{1}{z^6}=1$$ for all complex numbers $z$ with $|z|=1$, giving your answers in the form $\cos \phi +i\sin \phi$.