The problem in Exercice 1 involves proving by induction that for all integers $n \geq 2$, the term $u_n$ in the sequence defined by:

$u_2 = 3 \quad \text{and} \quad u_{n+1} = \frac{3u_n + 1}{u_n + 3}, \quad \text{for } n \geq 2$

can be expressed as:

$u_n = \frac{2^n + 2}{2^n - 2}.$

Plan for Inductive Proof:

1. Base Case: Verify that the formula holds for $n = 2$.

2. Inductive Hypothesis: Assume the formula holds for $n = k$, i.e.,

   $u_k = \frac{2^k + 2}{2^k - 2}.$

3. Inductive Step: Show that the formula holds for $n = k + 1$, i.e.,

   $u_{k+1} = \frac{2^{k+1} + 2}{2^{k+1} - 2}.$

We begin by verifying the base case and then proceed with the inductive steps.

Would you like a detailed step-by-step breakdown for each of these parts?