The director of a marine reserve counted 3000 cetaceans in this reserve on June 1st, 2017. He is concerned because he knows that the classification of the area as a ``marine reserve'' will not be renewed if the number of cetaceans in this reserve falls below 2000. A study allows him to develop a model according to which, each year:
between June 1st and October 31st, 80 cetaceans arrive in the marine reserve;
between November 1st and May 31st, the reserve experiences a decline of $5\%$ of its population compared to that of October 31st of the preceding year.
The evolution of the number of cetaceans is modelled by a sequence $(u_n)$. According to this model, for any natural number $n$, $u_n$ denotes the number of cetaceans on June 1st of the year $2017 + n$. We have $u_0 = 3000$.
Justify that $u_1 = 2926$.
Justify that, for any natural number $n$, $u_{n+1} = 0.95u_n + 76$.
Using a spreadsheet, the first 8 terms of the sequence $(u_n)$ were calculated. The director configured the cell format so that only numbers rounded to the nearest integer are displayed.
A
B
C
D
E
F
G
H
I
1
$n$
0
1
2
3
4
5
6
7
2
$u_n$
3000
2926
2856
2789
2725
2665
2608
2553
What formula can be entered in cell C2 to obtain, by copying to the right, the terms of the sequence $(u_n)$?
a. Prove that, for any natural number $n$, $u_n \geqslant 1520$. b. Prove that the sequence $(u_n)$ is decreasing. c. Justify that the sequence $(u_n)$ is convergent. We will not seek to find the value of the limit here.
We denote by $(v_n)$ the sequence defined by, for any natural number $n$, $v_n = u_n - 1520$. a. Prove that the sequence $(v_n)$ is a geometric sequence with ratio 0.95 and specify its first term. b. Deduce that, for any natural number $n$, $u_n = 1480 \times 0.95^n + 1520$. c. Determine the limit of the sequence $(u_n)$.
Copy and complete the following algorithm to determine the year from which the number of cetaceans present in the marine reserve will be less than 2000. $$\begin{array}{|l|}
\hline n \leftarrow 0 \\
u \leftarrow 3000 \\
\text{While } \ldots \\
\quad n \leftarrow \ldots \\
u \leftarrow \ldots \\
\text{End While}
\end{array}$$
The director of a marine reserve counted 3000 cetaceans in this reserve on June 1st, 2017. He is concerned because he knows that the classification of the area as a ``marine reserve'' will not be renewed if the number of cetaceans in this reserve falls below 2000.
A study allows him to develop a model according to which, each year:
\begin{itemize}
\item between June 1st and October 31st, 80 cetaceans arrive in the marine reserve;
\item between November 1st and May 31st, the reserve experiences a decline of $5\%$ of its population compared to that of October 31st of the preceding year.
\end{itemize}
The evolution of the number of cetaceans is modelled by a sequence $(u_n)$. According to this model, for any natural number $n$, $u_n$ denotes the number of cetaceans on June 1st of the year $2017 + n$. We have $u_0 = 3000$.
\begin{enumerate}
\item Justify that $u_1 = 2926$.
\item Justify that, for any natural number $n$, $u_{n+1} = 0.95u_n + 76$.
\item Using a spreadsheet, the first 8 terms of the sequence $(u_n)$ were calculated. The director configured the cell format so that only numbers rounded to the nearest integer are displayed.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | c | c | }
\hline
& A & B & C & D & E & F & G & H & I \\
\hline
1 & $n$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
\hline
2 & $u_n$ & 3000 & 2926 & 2856 & 2789 & 2725 & 2665 & 2608 & 2553 \\
\hline
\end{tabular}
\end{center}
What formula can be entered in cell C2 to obtain, by copying to the right, the terms of the sequence $(u_n)$?
\item a. Prove that, for any natural number $n$, $u_n \geqslant 1520$.\\
b. Prove that the sequence $(u_n)$ is decreasing.\\
c. Justify that the sequence $(u_n)$ is convergent. We will not seek to find the value of the limit here.
\item We denote by $(v_n)$ the sequence defined by, for any natural number $n$, $v_n = u_n - 1520$.\\
a. Prove that the sequence $(v_n)$ is a geometric sequence with ratio 0.95 and specify its first term.\\
b. Deduce that, for any natural number $n$, $u_n = 1480 \times 0.95^n + 1520$.\\
c. Determine the limit of the sequence $(u_n)$.
\item Copy and complete the following algorithm to determine the year from which the number of cetaceans present in the marine reserve will be less than 2000.
$$\begin{array}{|l|}
\hline n \leftarrow 0 \\
u \leftarrow 3000 \\
\text{While } \ldots \\
\quad n \leftarrow \ldots \\
u \leftarrow \ldots \\
\text{End While}
\end{array}$$
\end{enumerate}