The provided image contains a mathematical problem written in French. The statement is as follows:

Translation and Interpretation:

Given:

• An integer $n \geq 2$,
• $u_n = 9 \times 10^{n-2} - 10$.

Objective:

• Prove that $u_{n+1} = \dots$.

Let me analyze and solve the problem systematically:

1. General Pattern for $u_{n+1}$

The sequence $u_n$ is given by: $u_n = 9 \times 10^{n-2} - 10.$

To find $u_{n+1}$, substitute $n+1$ into the expression for $u_n$: $u_{n+1} = 9 \times 10^{(n+1) - 2} - 10.$ Simplify: $u_{n+1} = 9 \times 10^{n-1} - 10.$

2. Verify Consistency and Induction

This recurrence relation defines $u_{n+1}$ in terms of $n$. It appears the problem may aim to prove this relation using induction or demonstrate how $u_{n+1}$ satisfies the pattern.

Would you like me to continue with an induction proof or provide additional steps for clarity?

Follow-up Questions:

1. Would you like a step-by-step proof for this relation (e.g., using mathematical induction)?
2. Should I calculate a few terms of $u_n$ to validate the pattern?
3. Would you like an explanation of how this recurrence relates to sequences in general?
4. Should I interpret additional parts of the image for further problem context?
5. Would you like an explanation of why $9 \times 10^{n-2} - 10$ is significant in this sequence?

Tip:

When working with sequences, always calculate the first few terms to observe patterns before diving into formal proofs.