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The image contains three mathematical exercises asking to determine the variation of a sequence \( (u_n) \) for different given definitions. Here’s a breakdown of the three sequences:

### Exercise 1:
The sequence \( (u_n) \) is defined on \( \mathbb{N}^* \) (positive natural numbers) by:
\[
u_n = -3n^2 + 2n - 1
\]
You are asked to study the variation by analyzing \( u_{n+1} - u_n \), which involves computing the difference between consecutive terms of the sequence to determine whether the sequence is increasing or decreasing.

### Exercise 2:
The sequence \( (u_n) \) is defined on \( \mathbb{N} \) (natural numbers) by:
\[
u_n = ne^n
\]
You are asked to explain why \( u_n > 0 \) and study the ratio \( \frac{u_{n+1}}{u_n} \), which can help determine whether the sequence is growing exponentially or not.

### Exercise 3:
The sequence \( (u_n) \) is defined for all nonzero natural numbers \( n \) by:
\[
u_n = \frac{-n + 1}{n + 1}
\]
You are asked to study the variation of the function \( f: x \mapsto \frac{-x + 1}{x + 1} \) over the interval \( ]-1; +\infty[ \). This involves analyzing the behavior of the function and its derivative over this domain.

Would you like a detailed step-by-step solution for one or more of these exercises? Here are 5 related questions to consider:

1. How do you calculate the difference \( u_{n+1} - u_n \) for quadratic sequences?
2. What is the role of exponential growth in sequence 2's behavior?
3. How do you compute the ratio \( \frac{u_{n+1}}{u_n} \) for a given sequence?
4. What methods can be used to analyze the monotonicity of the function \( f(x) \) in Exercise 3?
5. How can you graphically represent the variations of these sequences?

**Tip:** When analyzing sequences, always check if there are recurring patterns in differences or ratios to determine whether a sequence is increasing, decreasing, or oscillating.

In each case, determine the nature of the variation of the sequence (u_n) defined by a given formula. Study the variation by calculating the first terms and analyzing the sequence.

Determine the variation of sequences defined by quadratic, exponential, and rational formulas. This problem includes studying the differences between consecutive terms, analyzing exponential growth, and investigating the behavior of rational functions over given domains. Suitable for advanced high school students.

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