<s>[INST] 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 ?
 [/INST] Edit: Found another solution, removed old answer (it was incorrect anyway)
You have $x+\frac{1}{x}=\frac{1+\sqrt{5}}{2}$. By simple algebraic manipulation you can get the
$$x^4-x^3+x^2-x+1 = 0$$
Now notice that $x^4 = x^3-x^2+x-1$ and multiplying both sides by $x$ you get  $x^5 = x^4-x^3+x^2-x=-1$.
Therefore
$$x^{2000}+\frac{1}{x^{2000}} = ({x^{5}})^{400}+\frac{1}{(x^{5})^{400}} = (-1)^{400}++\frac{1}{(-1)^{400}} = 1+1 = 2$$
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