This is a mathematical exercise involving sequences, functions, and recurrence relations. Here's a detailed breakdown of the tasks in the image:

1. Task 1: Variation Table for Function $f(x)$

   • The function $f(x) = \frac{3x}{1 + 2x}$ is defined on the set of positive real numbers.
   • You are asked to create a variation table for this function. This involves studying the monotonicity (whether the function is increasing or decreasing) and identifying its critical points by taking its derivative.

2. Task 2a: Recurrence Proof for Sequence $u_n$

   • The sequence $u_n$ is defined by the recurrence relation: $u_{n+1} = \frac{3u_n}{1 + 2u_n}, \quad u_0 = \frac{1}{2}.$
   • You need to prove, by induction, that $0 < u_n < 1$ for all $n \in \mathbb{N}$.

3. Task 2b: Study of Sequence $u_n$'s Monotonicity

   • Analyze whether the sequence $u_n$ is increasing or decreasing by comparing $u_{n+1}$ with $u_n$ and studying the behavior of the function $f(x)$.

4. Task 3: Convergence of Sequence $u_n$

   • Deduce from the previous steps that the sequence $u_n$ converges to a limit.

5. Task 4: Explicit Expression for $u_n$

   • Show that the general term of the sequence is given by: $u_n = \frac{3^n}{3^n + 1}.$
   • Then, use this expression to find the limit of the sequence as $n \to \infty$.

Now for Task 5, which builds on the previous work:

• Detailed Study of the Sequence $u_n$: This task involves fully analyzing the sequence $u_n$, particularly its behavior, convergence, and explicit expression. Here are some steps to expand on the previous tasks:
  • Complete the inductive proof of boundedness $0 < u_n < 1$.
  • Prove monotonicity using the recurrence relation.
  • Deduce that the sequence converges and calculate the limit.
  • Derive the explicit formula for the general term $u_n$ and prove it.

Would you like to work through one of these steps in detail or explore a specific task?

Here are 5 related questions for deeper understanding:

1. How do you calculate the derivative of $f(x) = \frac{3x}{1 + 2x}$ and use it to create the variation table?
2. How can you apply mathematical induction to prove that $0 < u_n < 1$?
3. What methods can you use to study the monotonicity of a recurrence relation?
4. How does proving that a sequence is bounded and monotonic lead to proving convergence?
5. How do you derive the explicit formula $u_n = \frac{3^n}{3^n + 1}$ from the recurrence relation?

Tip: In sequence problems, always check for boundedness and monotonicity to determine convergence effectively.