Recurrence relations

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

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?

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.

Can you find the value of this function involving algebraic fractions for x=2000?

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