Problem

Exercice 2:
On considère la suite définie par \( \left\{\begin{array}{l}u_{0}=1 \\ u_{n+1}=\sqrt{2+u_{n}^{2}}\end{array}\right. \). On pose \( v_{n}=u_{n}^{2} \)
1) Montrer que \( \left(v_{n}\right) \) est une suite arithmétique.
2) Exprimer \( v_{n} \) en fonction de \( n \). En déduire l'expression de \( u_{n} \) en fonction de \( r \).

Answer

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Answer

\(u_{n} = \sqrt{1 + 2n}\)

Steps

Step 1 ：\(v_{n+1} = u_{n+1}^2 = (\sqrt{2 + u_{n}^2})^2\)

Step 2 ：\(v_{n+1} = 2 + u_{n}^2 = v_n + 2\)

Step 3 ：\(v_{n} = 1 + 2n\)

Step 4 ：\(u_{n} = \sqrt{1 + 2n}\)

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