Problem #WSP-000111

Problems Algebra Sequences Recurrent relations Fibonacci numbers

Problem

$φ = \frac{1 + \sqrt{5}}{2}$ is the golden ratio. Using the fact that $φ^{2} = φ + 1$ , can you express $φ^{3}$ in the form $a φ + b$ , where $a$ and $b$ are positive integers?

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