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'Fibonacci Fashion' printed from https://nrich.maths.org/
$F_n={1\over\sqrt5}(\alpha^n-\beta^n)$ 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.