\textbf{(Candidates who have not followed the specialization course)}

A beekeeper studies the evolution of his bee population. At the beginning of his study, he estimates his bee population at 10000.\\
Each year, the beekeeper observes that he loses $20\%$ of the bees from the previous year.\\
He buys an identical number of new bees each year. We denote by $c$ this number expressed in tens of thousands.\\
We denote by $u _ { 0 }$ the number of bees, in tens of thousands, of this beekeeper at the beginning of the study.\\
For any non-zero natural number $n$, $u _ { n }$ denotes the number of bees, in tens of thousands, after the $n$-th year. Thus, we have

$$u _ { 0 } = 1 \quad \text { and, for any natural number } n , u _ { n + 1 } = 0.8 u _ { n } + c .$$

\section*{Part A}
We assume in this part only that $c = 1$.

\begin{enumerate}
  \item Conjecture the monotonicity and the limit of the sequence $\left( u _ { n } \right)$.
  \item Prove by induction that, for any natural number $n$, $u _ { n } = 5 - 4 \times 0.8 ^ { n }$.
  \item Verify the two conjectures established in question 1 by justifying your answer. Interpret these two results.
\end{enumerate}

\section*{Part B}
The beekeeper wants the number of bees to tend towards 100000.\\
We seek to determine the value of $c$ that allows reaching this objective.\\
We define the sequence $(v _ { n })$ by, for any natural number $n$, $v _ { n } = u _ { n } - 5 c$.

\begin{enumerate}
  \item Show that the sequence $\left( v _ { n } \right)$ is a geometric sequence and specify its common ratio and first term.
  \item Deduce an expression for the general term of the sequence $\left( v _ { n } \right)$ as a function of $n$.
  \item Determine the value of $c$ for the beekeeper to reach his objective.
\end{enumerate}