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The exercise you've provided defines a sequence \( (u_n) \) recursively, and the task seems to involve proving properties about it using mathematical induction.

Here's the given information extracted from the image:

- Initial term: \( u_1 = \frac{5}{2} \)
- Recursive formula for the sequence: 
  \[
  u_{n+1} = \frac{5u_n - 4}{2u_n - 1}, \quad \forall n \in \mathbb{N}^*
  \]
  
The first question in the exercise asks:
1) To prove by induction that for all \( n \in \mathbb{N}^* \),
  \[
  u_n = 2 + \frac{1}{3^{n-1}}
  \]

### To begin solving this using mathematical induction:

#### Base case (n = 1):
We are given that \( u_1 = \frac{5}{2} \), and we need to verify that this satisfies the formula for \( u_n \) when \( n = 1 \):
\[
u_1 = 2 + \frac{1}{3^{1-1}} = 2 + \frac{1}{1} = 3.
\]
Since \( u_1 \) is provided as \( \frac{5}{2} \), it looks like there might be a need to recheck the values or terms given.

Would you like to clarify or continue with the induction steps for this proof?

Prove by mathematical induction that for all n ∈ ℕ*, u_n = 2 + 1 / 3^(n-1), where the sequence (u_n) is defined as u_1 = 5/2 and u_{n+1} = (5u_n - 4) / (2u_n - 1).

Learn how to prove by induction that for all n ∈ ℕ*, the sequence u_n follows the formula u_n = 2 + 1 / 3^(n-1), starting from u_1 = 5/2 and with a recurrence relation u_{n+1} = (5u_n - 4) / (2u_n - 1). Suitable for advanced high school students.

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