Clean Minima

Algebra Level 3

Let \(\large \alpha\) and \(\large\beta\) be the roots of \(x^{2}-4(k^{2}-7k+8)x-7\).

If \(\large V_{n}=\alpha^{n}-\beta^{n}\) for \(n\geq1\).

Then Find the minimum possible value of

\[\large\dfrac{V_{101}-7V_{99}}{V_{100}}\]

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