\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
An association in a village of \(3,000\) inhabitants (constant population) studies the number of its volunteers.
In the year of its creation, there were \(20\) volunteers. Then, each year, it is estimated that \(25\pourcent\) of them leave the association, and \(5\pourcent\) of the inhabitants who were not volunteers the previous year become volunteers.
Let \(u_n\) be the number of volunteers and \(v_n\) the number of non-volunteers, \(n\) years after the creation.
  1. Give the values of \(u_0\) and \(v_0\).
  2. Give the value of \(u_n + v_n\) for all \(n \in \mathbb{N}\).
  3. Justify that for all \(n \in \mathbb{N}\), \(u_{n+1} = 0.75u_n + 0.05v_n\).
  4. Deduce that for all \(n \in \mathbb{N}\), \(u_{n+1} = 0.7u_n + 150\).
  5. Determine the expression of \(u_n\) in terms of \(n\).
  6. Determine the limit of the sequence \((u_n)\) and interpret it in the context of the problem.

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