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The text in the image describes a sequence \( u_n \) defined recursively by the relation:

\[
u_{n+1} = \frac{1}{2} u_n + 1 \quad \text{with} \quad u_1 = 4
\]

You've provided some instructions labeled from 1 to 3:

1. **Calculer les quatre premières termes de la suite (Calculate the first four terms of the sequence).**
2. **Expérimenter pour donner la valeur de \( u_n \) en fonction de \( n \) (Experiment to give the value of \( u_n \) as a function of \( n \)).**
3. **Déterminer une formule explicite pour \( u_n \) (Determine an explicit formula for \( u_n \)).**

Let's start by calculating the first four terms of the sequence \( u_n \).

### Calculating the Terms
Given \( u_1 = 4 \):

- \( u_2 = \frac{1}{2} u_1 + 1 = \frac{1}{2} \times 4 + 1 = 3 \)
- \( u_3 = \frac{1}{2} u_2 + 1 = \frac{1}{2} \times 3 + 1 = 2.5 \)
- \( u_

Soit (u_n) la suite définie par u_1 = 3 et, pour tout entier naturel n, u_{n+1} = 1/2 * u_n + 1. Calculer les quatre premières termes de la suite (u_n). Exprimer u_{n+1} en fonction de u_n et de n.

This problem explores a recursive sequence defined by u_1 = 3 and u_{n+1} = 1/2 * u_n + 1. Calculate the first four terms and express u_{n+1} in terms of u_n and n.

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