Recurrence relations

Investigate sequences given by $a_n = \frac{1+a_{n-1}}{a_{n-2}}$ for different choices of the first two terms. Make a conjecture about the behaviour of these sequences. Can you prove your conjecture?

You add 1 to the golden ratio to get its square. How do you find higher powers?

Each week a company produces X units and sells p per cent of its stock. How should the company plan its warehouse space?

A sequence of polynomials starts 0, 1 and each poly is given by combining the two polys in the sequence just before it. Investigate and prove results about the roots of the polys.