<s>[INST] Proving the form of a sequence's terms How do I go about attacking this problem and what is it asking?
Suppose that $\alpha^2 = \alpha + 1$ and suppose $F_n$ denotes the Fibonacci sequence.
Show that $\alpha^3 = 2\alpha +1, \alpha^4 = 3\alpha +2$ 
 [/INST] More explicitly,
since
$x^2 = x+1$,
$x^3 = x(x^2)
=x(x+1)
=x^2+x
=(x+1)+x
=2x+1
$.
Do the same for
$x^4$.
Essentially,
any power of $x$,
say $x^n$ with $n \ge 3$,
can be replaced
by lower powers of $x$
via
$x^n
=x^{n-2}x^2
=x^{n-2}(x+1)
=x^{n-1}+x^{n-2}
$.
Keep on applying this
until only linear and constant terms are left.
You will find an interesting formula
for $x^n$
in terms of $x$ and $1$.
</s>