The document you've uploaded presents an exercise involving a sequence of real numbers. It contains different sub-questions related to proving properties of the sequence and calculating sums and products.

Let's break down the main points:

1. Given Information:

   • A sequence $(u_n)$ is defined recursively:
     • $u_0 = -1$
     • $u_{n+1} = \frac{3u_n + 2}{u_n + 4}$
   • For all $n \in \mathbb{N}$, we are told that $-2 < u_n < 1$.

2. Questions:

   • (1) Prove that for all $n \in \mathbb{N}$, the condition $-2 < u_n < 1$ holds.
   • (2) (a) Prove that for all $n \in \mathbb{N}$, the expression for the difference between consecutive terms of the sequence is:
     $u_{n+1} - u_n = \frac{(1 - u_n)(u_n + 2)}{u_n + 4}$ (b) Deduce the monotonicity of the sequence $(u_n)$.
   • (3) A new sequence $(v_n)$ is defined by: $v_n = \frac{u_n - 1}{u_n + 2}$ (a) Prove that the sequence $(v_n)$ is geometric. (b) Express $v_n$ in terms of $n$, then express $u_n$ in terms of $n$.

3. Further Questions (4):

   • Sums and products are defined as: $S_n = v_0 + v_1 + \cdots + v_n, \quad P_n = v_0 \times v_1 \times \cdots \times v_n$ and $S = \frac{1}{u_0 + 2} + \frac{1}{u_1 + 2} + \cdots + \frac{1}{u_{10} + 2}$
   • (a) Calculate $S_n$, $P_n$, and $S$.
   • (b) Determine $m \in \mathbb{N}$ such that: $v_0 + v_1 + \cdots + v_m = \frac{-6090}{1875}$

Would you like to go over one of the specific parts in more detail, such as proving a formula or calculating one of the sums? Let me know how you'd like to proceed!