No use of any identities

Algebra Level 4

Let \(\alpha,\beta \) be the roots of the equation \(x^2-x-1=0\).

Let \(a_{2n}={\alpha}^{2n}+{\beta}^{2n} \) where \(n\geq1\) and \(n\) is a natural number.

What is the value of \(a_{20}+a_{22} \)?

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