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_