$F_n={1\over\sqrt5}(\alpha^n-\beta^n)$ where $\alpha$ and $\beta$ are solutions of the quadratic equation $x^2-x-1=0$ and $\alpha > \beta$ is the explicit formula for the $n$th Fibonacci number as given in this question.

The Golden Ratio is one of the roots of the quadratic equation and this explains the many connections between Fibonacci numbers and the Golden Ratio.

This problem complements the material in the article The Golden Ratio, Fibonacci Numbers and Continued Fractions

For a sequence of, mainly more elementary, problems on these
topics see

See this article if you want to know more about the method of proof by mathematical induction.