The exercise provided deals with sequences $(u_n)$ and $(v_n)$, defined recursively, and asks to perform various tasks with these sequences. Let's break down the problem and its steps.

Sequence $u_n$:

The sequence $(u_n)$ is defined by the recurrence relation:

$u_0 = \frac{1}{2}, \quad u_{n+1} = \frac{u_n + 3}{2u_n}$

You are asked to:

1. Graphical representation:

   • You need to plot the first 7 terms of the sequence $(u_n)$, from $u_0$ to $u_6$, and compare it with the graph of the function $f(x) = \frac{x + 3}{2x}$.

2. Calculate $u_1$ and $u_2$:

   • From the given recurrence relation, calculate the first few terms $u_1$ and $u_2$.
   • Determine whether the sequence $(u_n)$ is arithmetic or geometric.

3. Recurrence proof:

   • Prove by induction that $u_n > 0$ for all $n \in \mathbb{N}$.

4. Sequence $v_n$:

   • Define another sequence $(v_n)$ by the relation:

   $v_n = \frac{2u_n + 2}{2u_n - 3}$

   (a) Show that $(v_n)$ is geometric with a ratio of $-\frac{3}{2}$. Use this to find the general form for $v_n$:

   $v_n = \left( -\frac{3}{2} \right)^{n+1}$

   (b) Use this result to find the expression of $u_n$ as a function of $n$.

Next Steps:

Would you like me to proceed with the calculations for $u_1$, $u_2$, and verify the type of sequence? Or would you like to focus on a specific part of the problem, such as the graphical representation or the recurrence proof?

Here are 5 related questions:

1. How do we determine if a sequence is arithmetic or geometric?
2. What is the significance of proving that $u_n > 0$ for all $n$?
3. How do we solve recurrence relations to find a closed-form solution for sequences like $u_n$?
4. How does a geometric sequence differ from a general recursive sequence?
5. What is the relationship between the sequences $u_n$ and $v_n$?

Tip: Recurrence relations can often be transformed into simpler sequences using substitution, which can help reveal the underlying pattern or formula.