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The Golden Ratio is one of the roots of the equation $x^2-x-1=0$
and the $n$th Fibonacci number
is$F_n={1\over\sqrt5}(\alpha^n-\beta^n)$ where $\alpha$ and $\beta$
are solutions of the quadratic equation $x^2-x-1=0$ and $\alpha
> \beta$ hence the many connections between Fibonacci numbers
and the Golden Ratio.