Exercise 4 (5 points)

Candidates who have not followed the specialization course
A constant volume of $2200 \mathrm{~m}^{3}$ of water is distributed between two basins A and B. Basin A cools a machine. For reasons of thermal balance, a water current is created between the two basins using pumps. The exchanges between the two basins are modeled as follows:

• initially, basin A contains $800 \mathrm{~m}^{3}$ of water and basin B contains $1400 \mathrm{~m}^{3}$ of water;
• every day, 15\% of the volume of water present in basin B at the beginning of the day is transferred to basin A;
• every day, 10\% of the volume of water present in basin A at the beginning of the day is transferred to basin B. For every natural number $n$, we denote:
• $a_{n}$ the volume of water, expressed in $\mathrm{m}^{3}$, contained in basin A at the end of the $n$-th day of operation;
• $b_{n}$ the volume of water, expressed in $\mathrm{m}^{3}$, contained in basin B at the end of the $n$-th day of operation.

We therefore have $a_{0} = 800$ and $b_{0} = 1400$.

1. By what relation between $a_{n}$ and $b_{n}$ is the conservation of the total volume of water in the circuit expressed?
2. Justify that, for every natural number $n, a_{n+1} = \frac{3}{4} a_{n} + 330$.
3. The algorithm below makes it possible to determine the smallest value of $n$ from which $a_{n}$ is greater than or equal to 1100. Rewrite this algorithm by completing the missing parts.
   

   Variables	: $n$ is a natural number $a$ is a real number
   Initialization	: Assign to $n$ the value 0 Assign to $a$ the value 800
   Processing	: While $a < 1100$, do: Assign to $a$ the value . . . Assign to $n$ the value . . . End While
   Output	: Display $n$

   
   
4. For every natural number $n$, we denote $u_{n} = a_{n} - 1320$. a. Show that the sequence $(u_{n})$ is a geometric sequence and specify its first term and common ratio. b. Express $u_{n}$ as a function of $n$.

Deduce that, for every natural number $n, a_{n} = 1320 - 520 \times \left(\frac{3}{4}\right)^{n}$.
5. We seek to know if, on a given day, the two basins can have, to the nearest cubic meter, the same volume of water. Propose a method to answer this question.

A patient is given a medication by intravenous injection. The amount of medication in the blood decreases as a function of time. The purpose of the exercise is to study, for different hypotheses, the evolution of this amount minute by minute.

1. An injection of 10 mL of medication is performed at time 0. It is estimated that $20\%$ of the medication is eliminated per minute. For all natural integer $n$, we denote by $u_{n}$ the amount of medication, in mL, remaining in the blood after $n$ minutes. Thus $u_{0} = 10$. a. What is the nature of the sequence $\left(u_{n}\right)$? b. For all natural integer $n$, give the expression of $u_{n}$ as a function of $n$. c. After how much time does the amount of medication remaining in the blood become less than $1\%$ of the initial amount? Justify the answer.
2. A machine performs an injection of 10 mL of medication at time 0. It is estimated that $20\%$ of the medication is eliminated per minute. When the amount of medication falls below 5 mL, the machine reinjects 4 mL of product. After 15 minutes, the machine is stopped. For all natural integer $n$, we denote by $v_{n}$ the amount of medication, in mL, remaining in the blood at minute $n$. The following algorithm gives the amount of medication remaining minute by minute.
   

   \begin{tabular}{l} Variables:

   Initialization:

   Processing:

   &

   $n$ is a natural integer.

   $v$ is a real number.

   Assign to $v$ the value 10.

   For $n$ going from 1 to 15

   Assign to $v$ the value $0.8 \times v$.

   If $v < 5$ then assign to $v$ the value $v + 4$

   Display $v$.

   End of loop.

   \hline \end{tabular}
   a. Calculate the missing elements of the table below giving, rounded to $10^{-2}$ and for $n$ greater than or equal to 1, the amount of medication remaining minute by minute obtained with the algorithm.
   

   $n$	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
   $v_{n}$	10	8	6.4					8.15	6.52	5.21	8.17	6.54	5.23	8.18	6.55	5.24

   
   b. After 15 minutes, what total amount of medication has been injected into the body? c. We wish to program the machine so that it injects 2 mL of product when the amount of medication in the blood is less than or equal to 6 mL and that it stops after 30 minutes. Rewrite the previous algorithm by modifying it so that it displays the amount of medication, in mL, remaining in the blood minute by minute with this new protocol.
3. We program the machine so that:
   • at time 0, it injects 10 mL of medication,
   • every minute, it injects 1 mL of medication.
   It is estimated that $20\%$ of the medication present in the blood is eliminated per minute. For all natural integer $n$, we denote by $w_{n}$ the amount of medication, in mL, present in the blood of the patient after $n$ minutes. a. Justify that for all natural integer $n$, $w_{n+1} = 0.8w_{n} + 1$. b. For all natural integer $n$, we set $z_{n} = w_{n} - 5$. Prove that $(z_{n})$ is a geometric sequence whose ratio and first term we will specify. c. Deduce the expression of $w_{n}$ as a function of $n$. d. What is the limit of the sequence $\left(w_{n}\right)$? What interpretation can be given to this?

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

1. 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.
   

   Variables	$u$ and $n$ are numbers
   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

   \hline Output & Display .......... \hline \end{tabular}
   
2. a. Prove by induction that, for all natural integers $n$, $u _ { n } \geqslant 1000$. b. Prove that the sequence $( u _ { n } )$ is increasing.
3. 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 $( u _ { n } )$.

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 $t$.

1. 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 $t \to + \infty$.
2. Interpret the results of question 1 in the context of the problem.
3. 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.

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?

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

Part A

We assume in this part only that $c = 1$.

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

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 $n$, $v _ { n } = u _ { n } - 5 c$.

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

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 $u_0 = 3000$.

1. Justify that $u_1 = 2926$.
2. Justify that, for any natural number $n$, $u_{n+1} = 0.95u_n + 76$.
3. 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 $(u_n)$?
4. 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.
5. 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 $(u_n)$.
6. 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. $$\begin{array}{|l|} \hline n \leftarrow 0 \\ u \leftarrow 3000 \\ \text{While } \ldots \\ \quad n \leftarrow \ldots \\ u \leftarrow \ldots \\ \text{End While} \end{array}$$

In 2020, an influencer on social media has 1000 followers on her profile. The number of followers is modelled as follows: each year, she loses $10\%$ of her followers to which 250 new followers are added. For any natural integer $n$, we denote $u_{n}$ the number of followers on her profile in the year $(2020 + n)$, following this model. Thus $u_{0} = 1000$.

1. Calculate $u_{1}$.
2. Justify that for any natural integer $n$, $u_{n+1} = 0.9 u_{n} + 250$.
3. The Python function named ``suite'' is defined below. In the context of the exercise, interpret the value returned by suite(10).

\begin{verbatim} def suite(n) : u=1000 for i in range(n) : u=0.9*u+250 return u \end{verbatim}

1. a. Show, using a proof by induction, that for any natural integer $n$, $u_{n} \leqslant 2500$. b. Prove that the sequence $(u_{n})$ is increasing. c. Deduce from the previous questions that the sequence $(u_{n})$ is convergent.
2. Let $(v_{n})$ be the sequence defined by $v_{n} = u_{n} - 2500$ for any natural integer $n$. a. Show that the sequence $(v_{n})$ is a geometric sequence with common ratio 0.9 and initial term $v_{0} = -1500$. b. For any natural integer $n$, express $v_{n}$ as a function of $n$ and show that: $$u_{n} = -1500 \times 0.9^{n} + 2500$$ c. Determine the limit of the sequence $(u_{n})$ and interpret it in the context of the exercise.
3. Write a program that determines in which year the number of followers will exceed 2200. Determine this year.

Cécile has invited friends to lunch on her terrace. For dessert, she has planned an assortment of individual cakes that she bought frozen. She takes the cakes out of the freezer at $- 19 ^ { \circ } \mathrm { C }$ and brings them to the terrace where the temperature is $25 ^ { \circ } \mathrm { C }$. After 10 minutes, the temperature of the cakes is $1.3 ^ { \circ } \mathrm { C }$.

I- First model

We assume that the thawing rate is constant, that is, the temperature increase is the same minute after minute. According to this model, determine what the temperature of the cakes would be 25 minutes after they are taken out of the freezer. Does this model seem relevant?

II - Second model

We denote $T _ { n }$ the temperature of the cakes in degrees Celsius, after $n$ minutes following their removal from the freezer; thus $T _ { 0 } = - 19$. We assume that to model the evolution of temperature, we must have the following relation
$$\text { For all natural integers } n , T _ { n + 1 } - T _ { n } = - 0.06 \times \left( T _ { n } - 25 \right) \text {. }$$

1. Justify that, for all integers $n$, we have $T _ { n + 1 } = 0.94 T _ { n } + 1.5$
2. Calculate $T _ { 1 }$ and $T _ { 2 }$. Give values rounded to the nearest tenth.
3. Prove by induction that, for all natural integers $n$, we have $T _ { n } \leqslant 25$.

Returning to the situation studied, was this result foreseeable?
4. Study the direction of variation of the sequence $( T _ { n } )$.
5. Prove that the sequence $( T _ { n } )$ is convergent. 6. We set for all natural integers $n$, $U _ { n } = T _ { n } - 25$. a. Show that the sequence $( U _ { n } )$ is a geometric sequence and specify its common ratio and first term $U _ { 0 }$. b. Deduce that for all natural integers $n$, $T _ { n } = - 44 \times 0.94 ^ { n } + 25$. c. Deduce the limit of the sequence $( T _ { n } )$. Interpret this result in the context of the situation studied. 7. a. The manufacturer recommends consuming the cakes after half an hour at room temperature following their removal from the freezer. What is then the temperature reached by the cakes? Give a value rounded to the nearest integer. b. Cécile is a regular customer of these cakes, which she likes to enjoy while still fresh, at a temperature of $10 ^ { \circ } \mathrm { C }$. Give a range between two consecutive integers of the time in minutes after which Cécile should enjoy her cake. c. The following program, written in Python language, must return after its execution the smallest value of the integer $n$ for which $T _ { n } \geqslant 10$.
\begin{verbatim} def seuil() : n=0 T= while T T= n=n+1 return \end{verbatim}
Copy this program onto your paper and complete the incomplete lines so that the program returns the expected value.

A medication is administered to a patient intravenously.

Part A: discrete model of the medicinal quantity

After an initial injection of 1 mg of medication, the patient is placed on an infusion. It is estimated that, every 30 minutes, the patient's body eliminates 10\% of the quantity of medication present in the blood and receives an additional dose of 0.25 mg of the medicinal substance. We study the evolution of the quantity of medication in the blood with the following model: for any natural integer $n$, we denote by $u _ { n }$ the quantity, in mg, of medication in the patient's blood after $n$ periods of thirty minutes. We therefore have $u _ { 0 } = 1$.

1. Calculate the quantity of medication in the blood after half an hour.
2. Justify that, for any natural integer $n$, $u _ { n + 1 } = 0.9 u _ { n } + 0.25$.
3. a. Show by induction on $n$ that, for any natural integer $n$, $u _ { n } \leqslant u _ { n + 1 } < 5$. b. Deduce that the sequence $(u _ { n })$ is convergent.
4. It is estimated that the medication is truly effective when its quantity in the patient's blood is greater than or equal to 1.8 mg. a. Copy and complete the script written in Python language below so as to determine after how many periods of thirty minutes the medication begins to be truly effective. \begin{verbatim} def efficace(): u=1 n=0 while ......: u=...... n = n+1 return n \end{verbatim} b. What is the value returned by this script? Interpret this result in the context of the exercise.
5. Let $(v _ { n })$ be the sequence defined, for any natural integer $n$, by $v _ { n } = 2.5 - u _ { n }$. a. Show that $(v _ { n })$ is a geometric sequence and specify its common ratio and first term $(v _ { 0 })$. b. Show that, for any natural integer $n$, $u _ { n } = 2.5 - 1.5 \times 0.9 ^ { n }$. c. The medication becomes toxic when its quantity present in the patient's blood exceeds 3 mg. According to the chosen model, does the treatment present a risk for the patient? Justify.

Part B: continuous model of the medicinal quantity

After an initial injection of 1 mg of medication, the patient is placed on an infusion. The flow rate of the medicinal substance administered to the patient is 0.5 mg per hour. The quantity of medication in the patient's blood, as a function of time, is modeled by the function $f$, defined on $[ 0 ; + \infty [$, by $$f ( t ) = 2.5 - 1.5 \mathrm { e } ^ { - 0.2 t }$$ where $t$ denotes the duration of the infusion expressed in hours. We recall that this medication is truly effective when its quantity in the patient's blood is greater than or equal to 1.8 mg.

1. Is the medication truly effective after 3 hours 45 minutes?
2. According to this model, determine after how much time the medication becomes truly effective.
3. Compare the result obtained with that obtained in question 4. b. of the discrete model in Part A.

We study a group of 3000 athletes who practice either athletics in club A or basketball in club B. In 2023, club A has 1700 members and club B has 1300. We decide to model the number of members of club A and club B respectively by two sequences $(a_{n})$ and $(b_{n})$, where $n$ denotes the rank of the year starting from 2023. The year 2023 corresponds to rank 0. We then have $a_{0} = 1700$ and $b_{0} = 1300$. For our study, we make the following assumptions:

• during the study, no athlete leaves the group;
• each year, 15\% of the athletes in club A leave this club and join club B;
• each year, 10\% of the athletes in club B leave this club and join club A.

1. Calculate the number of members of each club in 2024.
2. For all natural integer $n$, determine a relation linking $a_{n}$ and $b_{n}$.
3. Show that the sequence $(a_{n})$ satisfies the following relation for all natural integer $n$: $$a_{n+1} = 0{,}75\, a_{n} + 300.$$
4. a. Prove by induction that for all natural integer $n$, we have: $$1200 \leqslant a_{n+1} \leqslant a_{n} \leqslant 1700.$$ b. Deduce that the sequence $(a_{n})$ converges.
5. Let $\left(v_{n}\right)$ be the sequence defined for all natural integer $n$ by $v_{n} = a_{n} - 1200$. a. Prove that the sequence $\left(v_{n}\right)$ is geometric. b. Express $v_{n}$ as a function of $n$. c. Deduce that for all natural integer $n$, $a_{n} = 500 \times 0{,}75^{n} + 1200$.
6. a. Determine the limit of the sequence $(a_{n})$. b. Interpret the result of the previous question in the context of the exercise.
7. a. Copy and complete the Python program below so that it returns the smallest value of $n$ from which the number of members of club A is strictly less than 1280. \begin{verbatim} def seuil() : n = 0 A = 1700 while... : n=n+1 A = ... return... \end{verbatim} b. Determine the value returned when the seuil function is called.

At the beginning of the experiment, we have a piece of 2 g of polonium. We know that 1 g of polonium contains $3 \times 10^{21}$ atomic nuclei. We assume that, after 24 hours, $0.5\%$ of the nuclei have disintegrated and that, to compensate for this loss, we then add $0.005\text{ g}$ of polonium. We model the situation using a sequence $\left(v_n\right)_{n \in \mathbb{N}}$; we denote $v_0$ the number of nuclei contained in the polonium at the beginning of the experiment. For $n \geqslant 1$, $v_n$ denotes the number of nuclei contained in the polonium after $n$ days have elapsed.

1. a. Verify that $v_0 = 6 \times 10^{21}$. b. Explain that, for every natural number $n$, we have $$v_{n+1} = 0{,}995\, v_n + 1{,}5 \times 10^{19}.$$
   
2. a. Prove, by induction on $n$, that $0 \leqslant v_{n+1} \leqslant v_n$. b. Deduce that the sequence $\left(v_n\right)_{n \in \mathbb{N}}$ is convergent.
   
3. We consider the sequence $\left(u_n\right)_{n \in \mathbb{N}}$ defined, for every natural number $n$, by: $$u_n = v_n - 3 \times 10^{21}.$$ a. Show that the sequence $\left(u_n\right)_{n \in \mathbb{N}}$ is geometric with common ratio 0.995. b. Deduce that, for every natural number $n$, $v_n = 3 \times 10^{21}\left(0{,}995^n + 1\right)$. c. Deduce the limit of the sequence $\left(v_n\right)_{n \in \mathbb{N}}$ and interpret the result in the context of the exercise.
   
4. Determine, by calculation, after how many days the number of polonium nuclei will be less than $4{,}5 \times 10^{21}$. Justify the answer.
   
5. We wish to have the list of terms of the sequence $\left(v_n\right)_{n \in \mathbb{N}}$. For this, we use a function called \texttt{noyaux} programmed in Python language and partially transcribed below. \begin{verbatim} def noyaux (n) : V =6*10**21 L=[V] for k in range (n) : V=... L.append(V) return L \end{verbatim} a. From reading the previous questions, propose two different solutions to complete line 5 of the \texttt{noyaux} function so that it answers the problem. b. For which value of the integer $n$ will the command \texttt{noyaux(n)} return the daily records of the number of nuclei contained in the polonium sample during 52 weeks of study?

A company has created a Frequently Asked Questions (``FAQ'') on its website. We study the number of questions asked there each month.

Part A: First modelling

In this part, we admit that, each month:

• $90\%$ of questions already asked the previous month are kept on the FAQ;
• 130 new questions are added to the FAQ.

During the first month, 300 questions were asked. To estimate the number of questions, in hundreds, present on the FAQ in the $n$-th month, we model the above situation using the sequence $(u_n)$ defined by: $$u_1 = 3 \text{ and, for all natural integer } n \geqslant 1, u_{n+1} = 0.9u_n + 1.3.$$

1. Calculate $u_2$ and $u_3$ and propose an interpretation in the context of the exercise.
2. Show by induction that for all natural integer $n \geqslant 1$: $$u_n = 13 - \frac{100}{9} \times 0.9^n.$$
3. Deduce that the sequence $(u_n)$ is increasing.
4. We consider the program opposite, written in Python language.
   Determine the value returned by entering \texttt{seuil(8.5)} and interpret it in the context of the exercise. \begin{verbatim} def seuil(p) : n=1 u=3 while u<=p : n=n+1 u=0.9*u+1.3 return n \end{verbatim}

Part B: Another modelling

In this part, we consider a second modelling using a new sequence $(v_n)$ defined for all natural integer $n \geqslant 1$ by: $$v_n = 9 - 6 \times \mathrm{e}^{-0.19 \times (n-1)}.$$ The term $v_n$ is an estimate of the number of questions, in hundreds, present in the $n$-th month on the FAQ.

1. Specify the values rounded to the nearest hundredth of $v_1$ and $v_2$.
2. Determine, by justifying the answer, the smallest value of $n$ such that $v_n > 8.5$.

Part C: Comparison of the two models

1. The company considers that it must modify the presentation of its site when more than 850 questions are present on the FAQ. Of these two modellings, which leads to making this modification the soonest? Justify your answer.
2. By justifying the answer, for which modelling is there the greatest number of questions on the FAQ in the long term?

Biologists are studying the evolution of an insect population in a botanical garden. At the beginning of the study, the population is 100,000 insects. To preserve the balance of the natural environment, the number of insects must not exceed 400,000.

Part A: Study of a first model in the laboratory

Observation of the evolution of these insect populations in the laboratory, in the absence of any predator, shows that the number of insects increases by $60\%$ each month. Taking this observation into account, biologists model the evolution of the insect population using a sequence $(u_n)$ where, for every natural number $n$, $u_n$ models the number of insects, expressed in millions, after $n$ months. We therefore have $u_0 = 0.1$.

1. Justify that for every natural number $n$: $u_n = 0.1 \times 1.6^n$.
2. Determine the limit of the sequence $(u_n)$.
3. By solving an inequality, determine the smallest natural number $n$ from which $u_n > 0.4$.
4. According to this model, would the balance of the natural environment be preserved? Justify your answer.

Part B: Study of a second model

Taking into account the constraints of the natural environment in which the insects evolve, biologists choose a new model. They model the number of insects using the sequence $(v_n)$, defined by: $$v_0 = 0.1 \text{ and, for every natural number } n, v_{n+1} = 1.6v_n - 1.6v_n^2,$$ where, for every natural number $n$, $v_n$ is the number of insects, expressed in millions, after $n$ months.

1. Determine the number of insects after one month.
2. We consider the function $f$ defined on the interval $\left[0; \frac{1}{2}\right]$ by $$f(x) = 1.6x - 1.6x^2.$$ a. Solve the equation $f(x) = x$. b. Show that the function $f$ is increasing on the interval $\left[0; \frac{1}{2}\right]$.
3. a. Show by induction that, for every natural number $n$, $0 \leqslant v_n \leqslant v_{n+1} \leqslant \frac{1}{2}$. b. Show that the sequence $(v_n)$ is convergent. We denote by $\ell$ the value of its limit. We admit that $\ell$ is a solution of the equation $f(x) = x$. c. Determine the value of $\ell$. According to this model, will the balance of the natural environment be preserved? Justify your answer.
4. The threshold function is given below, written in Python language. a. What do we observe if we enter \texttt{seuil(0.4)}? b. Determine the value returned by entering \texttt{seuil(0.35)}. Interpret this value in the context of the exercise. \begin{verbatim} def seuil(a) : v=0.1 n=0 while v

Exercise 1

We propose to compare the evolution of an animal population in two distinct environments A and B.
On January $1^{\text{st}}$ 2025, 6000 individuals are introduced into each of environments A and B.

Part A

In this part, we study the evolution of the population in environment A. We assume that in this environment, the evolution of the population is modelled by a geometric sequence $(u_n)$ with first term $u_0 = 6$ and common ratio 0.93. For every natural number $n$, $u_n$ represents the population on January $1^{\text{st}}$ of the year $2025 + n$, expressed in thousands of individuals.

1. Give, according to this model, the population on January $1^{\text{st}}$ 2026.
2. For every natural number $n$, express $u_n$ as a function of $n$.
3. Determine the limit of the sequence $(u_n)$.

Interpret this result in the context of the exercise.

Part B

In this part, we study the evolution of the population in environment B. We assume that in this environment, the evolution of the population is modelled by the sequence $(v_n)$ defined by
$$v_0 = 6 \text{ and for every natural number } n, v_{n+1} = -0.05 v_n^2 + 1.1 v_n.$$
For every natural number $n$, $v_n$ represents the population on January $1^{\text{st}}$ of the year $2025 + n$, expressed in thousands of individuals.

1. Give, according to this model, the population on January $1^{\text{st}}$ 2026.

Let $f$ be the function defined on the interval $[0; +\infty[$ by
$$f(x) = -0.05x^2 + 1.1x$$

1. Prove that the function $f$ is increasing on the interval $[0; 11]$.
2. Prove by induction that for every natural number $n$, we have $$2 \leqslant v_{n+1} \leqslant v_n \leqslant 6$$
3. Deduce that the sequence $(v_n)$ is convergent to a limit $\ell$.
4. a. Justify that the limit $\ell$ satisfies $f(\ell) = \ell$ then deduce the value of $\ell$. b. Interpret this result in the context of the exercise.

Part C

This part aims to compare the evolution of the population in the two environments.

1. By solving an inequality, determine the year from which the population of environment A will be strictly less than 3000 individuals.
2. Using a calculator, determine the year from which the population of environment B will be strictly less than 3000 individuals.
3. Justify that from a certain year onwards, the population of environment B will exceed the population of environment A.
4. Consider the Python program opposite. a. Copy and complete this program so that after execution, it displays the year from which the population of environment B is strictly greater than the population of environment A. b. Determine the year displayed after execution of the programme.

\begin{verbatim} n=0 u=6 v = 6 while...: u = ... v=... n = n+1 print (2025 + n) \end{verbatim}

Question 176
Um banco oferece uma aplicação com juros compostos de 1\% ao mês. Um cliente aplica R\$ 10 000,00. Após 2 meses, o montante obtido será de
(A) R\$ 10 100,00 (B) R\$ 10 200,00 (C) R\$ 10 201,00 (D) R\$ 10 210,00 (E) R\$ 10 220,00

To celebrate a city's anniversary, the city council organizes four consecutive days of cultural attractions. Experience from previous years shows that, from one day to the next, the number of visitors to the event is tripled. 345 visitors are expected for the first day of the event.
A possible representation of the expected number of participants for the last day is
(A) $3 \times 345$
(B) $(3 + 3 + 3) \times 345$
(C) $3^{3} \times 345$
(D) $3 \times 4 \times 345$
(E) $3^{4} \times 345$

A society where the proportion of the population aged 65 and over in the total population is 20\% or more is called a 'super-aged society'. In 2000, a certain country had a total population of 10 million and a population aged 65 and over of 500,000. Assuming that the total population increases by 0.3\% each year compared to the previous year and the population aged 65 and over increases by 4\% each year compared to the previous year, when is the first time a 'super-aged society' predicted to occur? (Given: $\log 1.003 = 0.0013$, $\log 1.04 = 0.0170$, $\log 2 = 0.3010$) [4 points]
(1) 2048--2050
(2) 2038--2040
(3) 2028--2030
(4) 2018--2020
(5) 2008--2010

A society where the proportion of the population aged 65 and over is 20\% or more of the total population is called a 'super-aged society'.
In the year 2000, a certain country had a total population of 10 million and a population aged 65 and over of 500,000. Assuming that the total population increases by 0.3\% each year compared to the previous year and the population aged 65 and over increases by 4\% each year compared to the previous year, when is the first time a 'super-aged society' predicted to occur? (Given: $\log 1.003 = 0.0013 , \log 1.04 = 0.0170 , \log 2 = 0.3010$) [4 points]
(1) 2048--2050
(2) 2038--2040
(3) 2028--2030
(4) 2018--2020
(5) 2008--2010

A father wants to distribute a certain sum of money between his daughter and son in such a way that if both of them invest their shares in the scheme that offers compound interest at $\frac { 25 } { 3 } \%$ per annum, for $t$ and $t + 2$ years respectively, then the two shares grow to become equal. If the son's share was rupees 4320, then the total money distributed by the father was
(A) rupees 9360
(B) rupees 9390
(C) rupees 16, 590
(D) rupees 16, 640.

In a census conducted on January 1, 2015, a city with a population of 810,000 had population counts on January 1 each year from 2016 to 2023. In each of the first four years after 2015, the population increased by a ratio of $\frac{1}{10}$ compared to the previous year, and in each of the following four years, the population increased by a ratio of $\frac{1}{11}$ compared to the previous year.
Accordingly, what was the population of this city in the census conducted on January 1, 2023?
A) $2^{20}$ B) $3^{13}$ C) $5^{9}$ D) $6^{8}$ E) $10^{6}$