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Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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.