Washington Crawford
08/09/2023 · High School
\begin{tabular}{|l|}\hline Considérons la suite \( \left(U_{n}\right) \) définie par: \( U_{0}=15 \) et \\ ( \( \forall n \in \mathbb{N})=U_{n+1}=\frac{7}{5} U_{n}-5 \) et \( \left(V_{n}\right) \) la suite définie \\ par: \( (\forall n \in \mathbb{N})=V_{n}=U_{n+1}-U_{n} \). \\ 1. Montrer que la suite \( \left(V_{n}\right) \) est géométrique dont \\ \( \quad \) on déterminera la raison. \\ 2. Déterminer \( V_{n} \) en fonction de \( n \). \\ 3. On pose \( S_{n}=\sum_{i=0}^{i=n} V_{i}=V_{0}+V_{1}+\cdots+V_{n} \). \\ 3.1. Montrer que \( (\forall n \in N): S_{n}=U_{n+1}-U_{0} \). \\ 3.2. En déduire \( U_{n} \) en fonction den. \\ \hline\end{tabular}
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La suite \( \left(V_{n}\right) \) est arithmétique avec \( V_{n} = 1 - 3n \). La somme \( S_{n} = U_{n+1} - 15 \), et \( U_{n} \) peut être exprimé en fonction de \( n \) comme \( U_{n} = 15 + \frac{(n+1)}{2} \cdot (2 - 3n) \).
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