### Golden Powers

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

### 2^n -n Numbers

Yatir from Israel wrote this article on numbers that can be written as $2^n-n$ where n is a positive integer.

### Poly Fibs

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.

# And So on - and on -and On

##### Age 16 to 18Challenge Level

The solution needs you to be systematic.

Start with $f_{0}$, then work out $f_{1}$, then work out $f_{2}$

\begin{eqnarray}f_{0}(2000)&=& \frac{1}{1-2000}\\ &=& \frac{1}{-1999}\end{eqnarray}