The problem in the image is written in Vietnamese, and it translates to:

Problem 18: Consider the sequence $(u_n)$ defined by: $u_0 \in \mathbb{R}, \; \forall n \in \mathbb{N}, \; u_{n+1} = \frac{u_n^2 + (-1)^n}{n+1}.$

We are asked to analyze or find some properties or values related to this sequence.

Analysis:

The given sequence $(u_n)$ is defined recursively:

$u_{n+1} = \frac{u_n^2 + (-1)^n}{n+1},$ with $u_0 \in \mathbb{R}$ (where $u_0$ is a real number) and $n \in \mathbb{N}$ (natural numbers, starting from 0).

Let's break down the definition:

1. Initial Condition: $u_0$ is any real number.

2. Recursive Formula: For every $n \in \mathbb{N}$, the next term $u_{n+1}$ is computed using the formula: $u_{n+1} = \frac{u_n^2 + (-1)^n}{n+1}.$

   • The term $u_n^2$ suggests that each term depends quadratically on the previous term.
   • The addition of $(-1)^n$ introduces alternating behavior, as it will add 1 when $n$ is even and subtract 1 when $n$ is odd.
   • The division by $n+1$ scales the result down by an increasing factor.

Observations:

• The sequence alternates in its behavior due to the $(-1)^n$ term.
• Understanding the growth or convergence of $u_n$ requires analyzing the behavior of the quadratic term and the alternating addition/subtraction.

To solve or explore further:

Would you like to find a specific term in the sequence, its general behavior, or determine if it converges or diverges?

Feel free to specify, and I can proceed with a more detailed solution or analysis.

Related Questions:

1. What is the value of $u_1$ given a specific $u_0$?
2. How does the sequence behave for small values of $u_0$, such as $u_0 = 0$ or $u_0 = 1$?
3. Does the sequence converge, and if so, to what value?
4. How does the alternating term $(-1)^n$ affect the sequence's overall behavior?
5. Can we find a closed-form expression for $u_n$?

Tip:

For recursive sequences, analyzing a few initial terms often helps identify patterns or trends that lead to a general formula or convergence behavior.