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### Exercise 2 Breakdown:

Given:
- The sequences \( (u_n) \) and \( (v_n) \) are defined by the following recurrence relations:
  \[
  u_{n+1} = u_n + \frac{v_n}{n}, \quad v_{n+1} = v_n + \frac{c}{n^2}, \quad \text{with } u_1 = v_1 = 1
  \]
  
#### 1. Write an expression for \( u_{n+1} - u_n \) and \( v_{n+1} - v_n \).

**Solution:**
- \( u_{n+1} - u_n = \frac{v_n}{n} \)
- \( v_{n+1} - v_n = \frac{c}{n^2} \)

These expressions represent the differences between consecutive terms in the sequences.

#### 2. Deduce \( v_n \) and show that \( v_n = 1 + c \sum_{k=1}^{n-1} \frac{1}{k^2} \).

**Solution:**
Starting from \( v_1 = 1 \) and using the recurrence relation:
\[
v_{n+1} = v_n + \frac{c}{n^2}
\]
we can express \( v_n \) as:
\[
v_n = v_1 + c \sum_{k=1}^{n-1} \frac{1}{k^2} = 1 + c \sum_{k=1}^{n-1} \frac{1}{k^2}
\]

#### 3. Deduce that \( u_n \) is equivalent to \( n v_n + S \) and find the value of \( S \).

**Solution:**
Given \( u_{n+1} = u_n + \frac{v_n}{n} \), summing over all terms gives:
\[
u_n = u_1 + \sum_{k=1}^{n-1} \frac{v_k}{k}
\]
Substituting \( v_k = 1 + c \sum_{j=1}^{k-1} \frac{1}{j^2} \) into this expression will help determine the form of \( u_n \) and the value of \( S \).

Would you like a deeper explanation for any of these steps or assistance with the remaining parts of the problem?

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Here are 5 related questions that expand on the topic:
1. How can you find the exact value of \( S \) in terms of the sequences \( u_n \) and \( v_n \)?
2. What happens to the sequences \( u_n \) and \( v_n \) as \( n \) approaches infinity?
3. Can you determine if the series \( \sum_{k=1}^{\infty} \frac{1}{k^2} \) converges or diverges?
4. How would you analyze the behavior of the sequence \( u_n - nv_n \) as \( n \) grows large?
5. What is the significance of the constant \( c \) in the recurrence relations, and how does it affect the sequences?

**Tip:** When working with recursive sequences, it's often helpful to look for patterns or sums that can simplify the recurrence relations.

Explore the analysis of recurrence relations in mathematical sequences. This problem delves into understanding recursive formulas, series summation, and applications of mathematical induction, suitable for preparatory level students.

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