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The exercise shown in the image asks to determine the nature of the variation of each sequence. Here’s a breakdown of the two parts provided in the image:

**Exercise 6:**
Determine the variation of each sequence.

1. The sequence \((u_n)\) is defined on \(\mathbb{N}\) by:  
   \( u_n = 3n - 4 \)

2. The sequence \((v_n)\) is defined on \(\mathbb{N}\) by:  
   \( v_n = \frac{n^2 + 2n - 3}{n - 5} \)

### Solutions:

**1. Sequence \((u_n) = 3n - 4\):**
This is an arithmetic sequence. The general term of an arithmetic sequence is of the form \( u_n = an + b \), where \(a\) is the common difference.

- The common difference here is 3, since \( u_{n+1} - u_n = 3(n+1) - 4 - (3n - 4) = 3 \).
- Therefore, the sequence is **strictly increasing** because the common difference is positive.

**2. Sequence \((v_n) = \frac{n^2 + 2n - 3}{n - 5}\):**
This is a rational sequence. To analyze its behavior, we can divide the numerator by the denominator using polynomial division:

\[
\frac{n^2 + 2n - 3}{n - 5} = n + 7 + \frac{32}{n - 5}
\]

- The term \( n + 7 \) increases as \(n\) increases, and the term \( \frac{32}{n - 5} \) decreases for \( n > 5 \), but more slowly as \(n\) grows.
- For large \(n\), \(v_n\) behaves approximately like \(n + 7\), so the sequence is **eventually increasing** for large \(n > 5\).

Do you want further details on the polynomial division or any other part of the solution? Here are five follow-up questions to deepen your understanding:

1. How do we calculate the common difference in an arithmetic sequence?
2. Why does the term \( \frac{32}{n - 5} \) decrease as \(n\) increases?
3. What happens to the sequence \(v_n\) when \( n = 5 \)?
4. How can we graph these sequences to visualize their variation?
5. What are the implications of the denominator in \(v_n\) for its domain of definition?

**Tip:** For rational sequences like \(v_n\), always check the denominator for undefined values (like \(n = 5\) here).

Determine the variation of each sequence.
1. The sequence (u_n) is defined on N by: u_n = 3n - 4.
2. The sequence (v_n) is defined on N by: v_n = (n^2 + 2n - 3) / (n - 5).

Explore the variation of sequences defined by arithmetic and rational functions. Learn how to identify increasing patterns and simplify complex expressions through polynomial division. Suitable for high school students.

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