Burton Higgins
02/01/2024 · Elementary School
Exercice A Soit la suite \( \left(u_{n}\right) \) définie par \( u_{0}=1 \) et pour tout entier \( n, u_{n+1}=\frac{2 u_{n}}{2+3 u_{n}} \). (1) Calculer les termes \( u_{1} \) et \( u_{2} \). (2) La suite \( \left(u_{n}\right) \) est-elle arithmétique ? géométrique ? (3) On admet que, pour tout \( n, u_{n} \) n'est pas nul. On pose \( v_{n}=1+\frac{2}{u_{n}} \). a) Calculer les trois premiers termes de \( \left(v_{n}\right) \). b) Déterminer la nature de \( \left(v_{n}\right) \). c) Exprimer \( v_{n} \) en fonction de \( n \). En déduire \( u_{n} \) en fonction de \( n \).
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(1) \( u_{1} = \frac{2}{5}, u_{2} = \frac{1}{4} \).
(2) La suite \( \left(u_{n}\right) \) n'est ni arithmétique ni géométrique.
(3) a) \( v_{0} = 3, v_{1} = 6, v_{2} = 9 \). b) La suite \( \left(v_{n}\right) \) est arithmétique. c) \( u_{n} = \frac{2}{3n + 2} \).
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