If $x+\frac{1}{x}=\frac{1+\sqrt{5}}{2}$ then $x^{2000}+\frac{1}{x^{2000}}= $? If $x+\frac{1}{x}=\frac{1+\sqrt{5}}{2}$ then
$$x^{2000}+\frac{1}{x^{2000}}=?$$
My try:
$$\left(x^{1000}\right)^2+\left(\frac{1}{x^{1000}}\right)^2=\left(x^{1000}+\frac{1}{x^{1000}}\right)^2-2$$
Continuation ?

Let $\psi = \frac{1+\sqrt{5}}{2} $
$$x + \frac{1}{x} = \psi$$
$$x^2 + 1 = \psi x $$
$$x^2 - \psi x + 1 = 0$$
Use quadratic formula here to solve for $x$. Then plug that into the given expression.
