The image contains a mathematics exercise related to a recursive sequence, where the terms $U_n$ are defined as:

$U_0 = 1 \quad \text{and} \quad U_{n+1} = \frac{5U_n - 3}{U_n + 1}, \quad \forall n \in \mathbb{N}.$

Here are the tasks you're asked to perform:

1. Calculate $U_1$ and $U_2$.
2. Show by induction that $U_n$ is between 1 and 3 for all $n \in \mathbb{N}$.
3. a) Show that $U_{n+1} - U_n = -\frac{(U_n - 1)(U_n - 3)}{U_n + 1}.$ b) Determine the monotonicity of the sequence $(U_n)$.
4. For each $n \in \mathbb{N},$ define a new sequence $V_n = U_n - 3$. a) Show that $V_n$ is a geometric sequence with a common ratio. b) Express $V_n$ and then $U_n$ in terms of $n$. c) Calculate the sum $V_1 + V_2 + V_3 + \dots + V_n$.

Would you like help solving any of these tasks, or do you have specific questions about one of them?