The periodic sequence $w_n$ is defined by

where $a,b\ne 0$.

1. Show that $w_3$ and $w_4$ are given by

   $$\begin{array}{rl}{w}_3& =\frac{a^2+ab+b}{a+b}\\ w_4& =\frac{a^3+a^{2}b+2ab+b^2}{a^2+ab+b}\end{array}$$

2. Explain why it is not possible for the period of the sequence to be $2$.

3. Prove that, when $b=-a^2$, the period of the sequence is $3$.

For (b), consider both

and

You should be able to show that the period can't be $2$ because, if $w_3=w_1$ then the period must be $1$.

For (c), it is not sufficient to only show that

gives us $w_4=w_1$. You must also show that the period of the sequence is exactly $3$ (i.e. it is neither $1$ nor $2$).