algèbre lycée recueilchapitre 1exercice 1.1fibonacci

Binet's formula

Recueil COMIMa — Techniques de résolution de problèmes · 2026 · Madagascar · ★★★★★

Statement

Démontrer la formule de Binet pour les nombres de Fibonacci : \[ F_n = \frac{\alpha^n - \beta^n}{\sqrt{5}}, \quad \text{où } \alpha = \frac{1 + \sqrt{5}}{2} \text{ et } \beta = \frac{1 - \sqrt{5}}{2}. \]

Indication : Utiliser la relation de récurrence $F_{n+2} = F_{n+1} + F_n$.

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Origin : Recueil COMIMa — Techniques de résolution de problèmes · 2026 · Madagascar

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