Consider the sequence $\left( u _ { n } \right)$ defined by $u _ { 0 } = 10000$ and for every natural number $n$ :

$$u _ { n + 1 } = 0,95 u _ { n } + 200 .$$

\begin{enumerate}
  \item Calculate $u _ { 1 }$ and verify that $u _ { 2 } = 9415$.
  \item a. Prove, using proof by induction, that for every natural number $n$ :
$$u _ { n } > 4000$$
b. It is admitted that the sequence $(u _ { n })$ is decreasing. Justify that it converges.
  \item For every natural number $n$, consider the sequence $\left( v _ { n } \right)$ defined by: $v _ { n } = u _ { n } - 4000$.\\
a. Calculate $v _ { 0 }$.\\
b. Prove that the sequence $(v _ { n })$ is geometric with common ratio equal to 0.95.\\
c. Deduce that for every natural number $n$ :
$$u _ { n } = 4000 + 6000 \times 0,95 ^ { n }$$
d. What is the limit of the sequence $\left( u _ { n } \right)$? Justify your answer.
  \item In 2020, an animal species numbered 10000 individuals. The evolution observed in previous years leads to the estimate that from 2021 onwards, this population will decrease by $5\%$ at the beginning of each year.\\
To slow down this decline, it was decided to reintroduce 200 individuals at the end of each year, starting from 2021.\\
A representative of an association supporting this strategy claims that: ``the species should not become extinct, but unfortunately, we will not prevent a loss of more than half the population''.\\
What do you think of this statement? Justify your answer.
\end{enumerate}