### 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.

### And So on - and on -and On

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

# Poly Fibs

##### Stage: 5 Challenge Level:

Consider the sequence of polynomials given by $P_{n+2}(x)=xP_{n+1}(x)-P_n(x)$ where $P_0(x)=0$ and $P_1(x)=1$

(i) Show that every root of $P_3$ is a root of $P_6$.

(ii) Show that every root of $P_4$ is a root of $P_8$.

(iii) Show that every root of $P_5$ is a root of $P_{10}$.

You can do this by finding the polynomials and then finding their roots (maybe using a computer), but try to find another way to get this result without finding the roots of the polynomials.

One of the skills of a research mathematician is making conjectures about results that no-one has thought of and that turn out to be provable. In this problem there is a conjecture about a general result which you may be able to make quite easily although the proof is well beyond the scope of school mathematics. Go on learning mathematics and in a few years you will be able to prove it.
The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities can be found here.
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