A company produces bacteria for industry. In the laboratory, it was measured that, in an appropriate nutrient medium, the mass of these bacteria, measured in grams, increases by $20\%$ in one day. The company implements the following industrial process. In a vat of nutrient medium, 1 kg of bacteria is initially introduced. Then, each day, at a fixed time, the nutrient medium in the vat is replaced. During this operation, 100 g of bacteria are lost. The company's objective is to produce 30 kg of bacteria.

\textbf{Part A: first model - with a sequence}

The evolution of the bacterial population in the vat is modeled by the sequence $(u _ { n })$ defined as follows:

$$u _ { 0 } = 1000 \text{ and, for all natural integers } n , u _ { n + 1 } = 1.2 u _ { n } - 100 .$$

\begin{enumerate}
  \item a. Explain how this model corresponds to the situation described in the problem. You will specify in particular what $u _ { n }$ represents.\\
b. The company wants to know after how many days the mass of bacteria will exceed 30 kg. Using a calculator, give the answer to this problem.\\
c. We can also use the following algorithm to answer the problem posed in the previous question. Copy and complete this algorithm.

\begin{center}
\begin{tabular}{|l|l|}
\hline
Variables & $u$ and $n$ are numbers \\
\hline
Processing & \begin{tabular}{l}
$u$ takes the value 1000 \\
$n$ takes the value 0 \\
While $\_\_\_\_$ do \\
$u$ takes the value $\_\_\_\_$ $n$ takes the value $n + 1$ \\
End While \\
\end{tabular} \\
\hline
Output & Display .......... \\
\hline
\end{tabular}
\end{center}

  \item a. Prove by induction that, for all natural integers $n$, $u _ { n } \geqslant 1000$.\\
b. Prove that the sequence $( u _ { n } )$ is increasing.
  \item We define the sequence $( v _ { n } )$ by: for all natural integers $n$, $v _ { n } = u _ { n } - 500$.\\
a. Prove that the sequence $( v _ { n } )$ is a geometric sequence.\\
b. Express $v _ { n }$, then $u _ { n }$, as a function of $n$.\\
c. Determine the limit of the sequence $( u _ { n } )$.
\end{enumerate}

\textbf{Part B: second model - with a function}

It is observed that in practice, the mass of bacteria in the vat will never exceed 50 kg. This leads to studying a second model in which the mass of bacteria is modeled by the function $f$ defined on $[ 0 ; +\infty[$ by:

$$f ( t ) = \frac { 50 } { 1 + 49 \mathrm { e } ^ { - 0.2 t } }$$

where $t$ represents time expressed in days and where $f ( t )$ represents the mass, expressed in kg, of bacteria at time $t$.

\begin{enumerate}
  \item a. Calculate $f ( 0 )$.\\
b. Prove that, for all real $t \geqslant 0$, $f ( t ) < 50$.\\
c. Study the monotonicity of the function $f$.\\
d. Determine the limit of the function $f$ as $t \to + \infty$.
  \item Interpret the results of question 1 in the context of the problem.
  \item Using this model, we seek to determine after how many days the mass of bacteria will exceed 30 kg. Solve the inequality with unknown $t$: $f ( t ) > 30$. Deduce the answer to the problem.
\end{enumerate}

\textbf{Part C: quality control}

Bacteria can be of two types: type A, which effectively produces a protein useful to industry, and type B, which does not produce it and is therefore commercially useless. The company claims that $80\%$ of the bacteria produced are of type A. To verify this claim, a laboratory analyzes a random sample of 200 bacteria at the end of production. The analysis shows that 146 of them are of type A. Should the company's claim be questioned?