Ellis Lambert
10/29/2023 · Senior High School
Exercice B On définit une suite \( \left(u_{n}\right) \) par \( \left\{u_{n+1}=\frac{1}{2} u_{n}+2 n-i\right. \) pour tout \( n \in \mathbb{N} \). (1) Calculer les premiers termes de la suite \( \left(u_{n}\right) \). Que peut-on conjecturer concernant sa nature et son sens de variation? (2) On pose \( v_{n}=u_{n}-4 n+10 \). a) Montrer que \( \left(v_{n}\right) \) est une suite géométrique que l'on caractérisera. b) En déduire l'expression de \( v_{n} \) en fonction de \( n \) puis celle de \( u_{n} \) en fonction de \( n \). c) On pose \( s_{n}=\sum u_{k}=u_{0}+u_{1}+\ldots+u_{n} \). Donner l'expression de \( s_{n} \) en
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### (1) Calcul des premiers termes de la suite \( \left(u_{n}\right) \)
- Pour \( n = 0 \) : \( u_1 = \frac{1}{2} u_0 - i \)
- Pour \( n = 1 \) : \( u_2 = \frac{1}{4} u_0 + 2 - \frac{3}{2} i \)
- Pour \( n = 2 \) : \( u_3 = \frac{1}{8} u_0 + 5 - \frac{7}{4} i \)
### (2) Transformation de la suite
#### a) Montrer que \( \left(v_n\right) \) est une suite géométrique
- \( v_{n+1} = \frac{1}{2}v_n + 6 - i \)
#### b) Expression de \( v_n \) en fonction de \( n \)
- \( v_n = (u_0 - 10) \left(\frac{1}{2}\right)^n + \text{(termes constants)} \)
#### c) Expression de \( s_n = \sum_{k=0}^{n} u_k \)
- \( s_n = \sum_{k=0}^{n} v_k + 2n(n + 1) - 10(n + 1) \)
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