- A sequence of non-zero real numbers \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by
$$a _ { n + 1 } = p + \frac { q } { a _ { n } } \quad n \in \mathbb { N }$$
where \(p\) and \(q\) are real numbers with \(q \neq 0\)
It is known that
- one of the terms of this sequence is a
- the sequence is periodic
- Determine an equation for \(q\), in terms of \(p\) and \(a\), such that the sequence is constant (of period/order one).
- Determine the value of \(p\) that is necessary for the sequence to be of period/order 2.
- Give an example of a sequence that satisfies the condition in part (b), but is not of period/order 2.
- Determine an equation for \(q\), in terms of \(p\) only, such that the sequence has period/order 4.