\textbf{Exercise A} (Main topics: Sequences, Differential equations)

In this exercise, we are interested in the growth of Moso bamboo with maximum height 20 meters. Ludwig von Bertalanffy's growth model assumes that the growth rate for such bamboo is proportional to the difference between its height and the maximum height.

\section*{Part I: discrete model}
In this part, we observe a bamboo with initial height 1 meter. For every natural integer $n$, we denote $u_n$ the height, in meters, of the bamboo $n$ days after the start of observation. Thus $u_0 = 1$. Von Bertalanffy's model for bamboo growth between two consecutive days is expressed by the equality:
$$u_{n+1} = u_n + 0.05\left(20 - u_n\right) \text{ for every natural integer } n.$$

\begin{enumerate}
  \item Verify that $u_1 = 1.95$.
  \item a. Show that for every natural integer $n$, $u_{n+1} = 0.95 u_n + 1$.\\
b. We set for every natural integer $n$, $v_n = 20 - u_n$. Prove that the sequence $(v_n)$ is a geometric sequence and specify its initial term $v_0$ and its common ratio.\\
c. Deduce that, for every natural integer $n$, $u_n = 20 - 19 \times 0.95^n$.
  \item Determine the limit of the sequence $(u_n)$.
\end{enumerate}

\section*{Part II: continuous model}
In this part, we wish to model the height of the same Moso bamboo by a function giving its height, in meters, as a function of time $t$ expressed in days. According to von Bertalanffy's model, this function is a solution of the differential equation
$$(E) \quad y^{\prime} = 0.05(20 - y)$$
where $y$ denotes a function of the variable $t$, defined and differentiable on $[0; +\infty[$ and $y^{\prime}$ denotes its derivative function.\\
Let the function $L$ defined on the interval $[0; +\infty[$ by
$$L(t) = 20 - 19\mathrm{e}^{-0.05t}$$

\begin{enumerate}
  \item Verify that the function $L$ is a solution of $(E)$ and that we also have $L(0) = 1$.
  \item We take this function $L$ as our model and we admit that, if we denote $L^{\prime}$ its derivative function, $L^{\prime}(t)$ represents the growth rate of the bamboo at time $t$.\\
a. Compare $L^{\prime}(0)$ and $L^{\prime}(5)$.\\
b. Calculate the limit of the derivative function $L^{\prime}$ at $+\infty$. Is this result consistent with the description of the growth model presented at the beginning of the exercise?
\end{enumerate}