We define, for every natural integer $n$, the complex numbers $z$ by:
$$\begin{cases} z_{0} & = 16 \\ z_{n+1} & = \frac{1 + \mathrm{i}}{2} z_{n}, \text{ for every natural integer } n. \end{cases}$$
We denote $r_{n}$ the modulus of the complex number $z_{n}$: $r_{n} = |z_{n}|$. In the plane equipped with a direct orthonormal coordinate system with origin O, we consider the points $A_{n}$ with affixes $z_{n}$.

1. a) Calculate $z_{1}, z_{2}$ and $z_{3}$. b) Plot the points $A_{1}$ and $A_{2}$ on the graph given in the appendix, to be returned with your answer sheet. c) Write the complex number $\frac{1 + \mathrm{i}}{2}$ in trigonometric form. d) Prove that the triangle $\mathrm{OA}_{0}A_{1}$ is isosceles right-angled at $A_{1}$.
2. Prove that the sequence $(r_{n})$ is geometric, with common ratio $\frac{\sqrt{2}}{2}$.

Is the sequence $(r_{n})$ convergent? Interpret the previous result geometrically. We denote $L_{n}$ the length of the broken line connecting point $A_{0}$ to point $A_{n}$ passing successively through points $A_{1}, A_{2}, A_{3}$, etc. Thus $L_{n} = \sum_{i=0}^{n-1} A_{i}A_{i+1} = A_{0}A_{1} + A_{1}A_{2} + \ldots + A_{n-1}A_{n}$.
3. a) Prove that for every natural integer $n$: $A_{n}A_{n+1} = r_{n+1}$. b) Give an expression for $L_{n}$ as a function of $n$. c) Determine the possible limit of the sequence $(L_{n})$.

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?

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 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.

Consider the sequence $(u_n)$ defined by:
$$\left\{ \begin{aligned} u _ { 0 } & = 1 \text{ and, for every natural number } n, \\ u _ { n + 1 } & = \left( \frac { n + 1 } { 2 n + 4 } \right) u _ { n } . \end{aligned} \right.$$
Define the sequence $(v_n)$ by: for every natural number $n$, $v_n = (n + 1) u_n$.

1. The spreadsheet below presents the values of the first terms of the sequences $(u_n)$ and $(v_n)$, rounded to the hundred-thousandth. What formula, then extended downward, can be written in cell B3 of the spreadsheet to obtain the successive terms of $(u_n)$?
   

   	A	B	C
   1	$n$	$u_n$	$v_n$
   2	0	1.00000	1.00000
   3	1	0.25000	0.50000
   4	2	0.08333	0.25000
   5	3	0.03125	0.12500
   6	4	0.01250	0.06250
   7	5	0.00521	0.03125
   8	6	0.00223	0.01563
   9	7	0.00098	0.00781
   10	8	0.00043	0.00391
   11	9	0.00020	0.00195

   
   
2. a. Conjecture the expression of $v_n$ as a function of $n$. b. Prove this conjecture.
3. Determine the limit of the sequence $(u_n)$.

For $n \geqslant 1$, we set $p_n = P(A_n)$ with $p_1 = 1$ and $p_{n+1} = 0{,}5\, p_n + 0{,}4$. We set for all integer $n \geqslant 1$: $v_n = p_n - 0{,}8$. a. Prove that $(v_n)$ is a geometric sequence and give its first term $v_1$ and common ratio. b. Express $v_n$ as a function of $n$. Deduce that, for all $n \geqslant 1$, $p_n = 0{,}8 + 0{,}2 \times 0{,}5^{n-1}$. c. Determine the limit of the sequence $(p_n)$.

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. $$\begin{array}{|l|} \hline n \leftarrow 0 \\ u \leftarrow 3000 \\ \text{While } \ldots \\ \quad n \leftarrow \ldots \\ u \leftarrow \ldots \\ \text{End While} \end{array}$$

(Candidates who have not followed the specialisation course)
A computer game of chance is set up as follows:

• If the player wins a game, the probability that he wins the next game is $\frac{1}{4}$;
• If the player loses a game, the probability that he loses the next game is $\frac{1}{2}$;
• The probability of winning the first game is $\frac{1}{4}$.

For every non-zero natural number $n$, we denote by $G_{n}$ the event ``the $n^{\mathrm{th}}$ game is won'' and we denote by $p_{n}$ the probability of this event. We thus have $p_{1} = \frac{1}{4}$.

1. Show that $p_{2} = \frac{7}{16}$.
2. Show that, for every non-zero natural number $n$, $p_{n+1} = -\frac{1}{4} p_{n} + \frac{1}{2}$.
3. We thus obtain the first values of $p_{n}$:
   

   $n$	1	2	3	4	5	6	7
   $p_{n}$	0,25	0,4375	0,3906	0,4023	0,3994	0,4001	0,3999

   
   What conjecture can be made?
4. We define, for every non-zero natural number $n$, the sequence $(u_{n})$ by $u_{n} = p_{n} - \frac{2}{5}$. a. Prove that the sequence $(u_{n})$ is a geometric sequence and specify its common ratio. b. Deduce that, for every non-zero natural number $n$, $p_{n} = \frac{2}{5} - \frac{3}{20}\left(-\frac{1}{4}\right)^{n-1}$. c. Does the sequence $(p_{n})$ converge? Interpret this result.

Newton's law of cooling states that the rate of change of the temperature of a body is proportional to the difference between the temperature of this body and that of the surrounding environment.
A cup of coffee is served at an initial temperature of $80^{\circ}\mathrm{C}$ in an environment whose temperature, expressed in degrees Celsius, assumed to be constant, is denoted $M$.
In this part, for any natural number $n$, we denote $T_n$ the temperature of the coffee at instant $n$, with $T_n$ expressed in degrees Celsius and $n$ in minutes. Thus $T_0 = 80$.
We model Newton's law between any two consecutive minutes $n$ and $n+1$ by the equality: $$T_{n+1} - T_n = k(T_n - M)$$ where $k$ is a real constant.
In the rest of part A, we choose $M = 10$ and $k = -0{,}2$. Thus, for any natural number $n$, we have: $T_{n+1} - T_n = -0{,}2(T_n - 10)$.

1. Based on the context, can we conjecture the direction of variation of the sequence $(T_n)$?
2. Show that for any natural number $n$: $T_{n+1} = 0{,}8T_n + 2$.
3. We set, for any natural number $n$: $u_n = T_n - 10$. a. Show that $(u_n)$ is a geometric sequence. Specify its common ratio and its first term $u_0$. b. Show that, for any natural number $n$, we have: $T_n = 70 \times 0{,}8^n + 10$. c. Determine the limit of the sequence $(T_n)$.
4. Consider the following algorithm: \begin{verbatim} While $T \geqslant 40$ $T \leftarrow 0,8T + 2$ $n \leftarrow n + 1$ End While \end{verbatim} a. Initially, we assign the value 80 to the variable $T$ and the value 0 to the variable $n$. What numerical value does the variable $n$ contain at the end of the algorithm's execution? b. Interpret this value in the context of the exercise.

In May 2020, a company chose to develop telework. It proposed to its 5000 employees in France to choose between telework and working at the company's premises. In May 2020, only 200 of them chose telework. Each month, since the implementation of this measure, $85\%$ of those who had chosen telework the previous month choose to continue, and each month, 450 additional employees choose telework. The number of company employees working from home is modeled by the sequence $(a_n)$. The term $a_n$ designates an estimate of the number of employees working from home in the $n$-th month after May 2020. Thus $a_0 = 200$.
Part A:

1. Calculate $a_1$.
2. Justify that for every natural number $n$, $a_{n+1} = 0.85a_n + 450$.
3. Consider the sequence $(v_n)$ defined for every natural number $n$ by: $v_n = a_n - 3000$. a. Prove that the sequence $(v_n)$ is a geometric sequence with common ratio 0.85. b. Express $v_n$ as a function of $n$ for every natural number $n$. c. Deduce that, for every natural number $n$, $a_n = -2800 \times 0.85^n + 3000$.
4. Determine the number of months after which the number of teleworkers will be strictly greater than 2500, after the implementation of this measure in the company.

Part B: The company's managers modeled the number of employees satisfied with this system using the sequence $(u_n)$ defined by $u_0 = 1$ and, for every natural number $n$, $$u_{n+1} = \frac{5u_n + 4}{u_n + 2}$$ where $u_n$ denotes the number of thousands of employees satisfied with this new measure after $n$ months following May 2020.

1. Prove that the function $f$ defined for all $x \in [0;+\infty[$ by $f(x) = \dfrac{5x+4}{x+2}$ is strictly increasing on $[0;+\infty[$.
2. a. Prove by induction that for every natural number $n$: $$0 \leqslant u_n \leqslant u_{n+1} \leqslant 4.$$ b. Justify that the sequence $(u_n)$ is convergent.
3. We admit that for every natural number $n$, $$0 \leqslant 4 - u_n \leqslant 3 \times \left(\frac{1}{2}\right)^n.$$ Deduce the limit of the sequence $(u_n)$ and interpret it in the context of the modeling.

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}

\setcounter{enumi}{3}
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.

Consider the sequence $\left( u _ { n } \right)$ defined by $u _ { 0 } = 10000$ and for every natural number $n$ :
$$u _ { n + 1 } = 0,95 u _ { n } + 200 .$$

1. Calculate $u _ { 1 }$ and verify that $u _ { 2 } = 9415$.
2. a. Prove, using proof by induction, that for every natural number $n$ : $$u _ { n } > 4000$$ b. It is admitted that the sequence $(u _ { n })$ is decreasing. Justify that it converges.
3. For every natural number $n$, consider the sequence $\left( v _ { n } \right)$ defined by: $v _ { n } = u _ { n } - 4000$. a. Calculate $v _ { 0 }$. b. Prove that the sequence $(v _ { n })$ is geometric with common ratio equal to 0.95. c. Deduce that for every natural number $n$ : $$u _ { n } = 4000 + 6000 \times 0,95 ^ { n }$$ d. What is the limit of the sequence $\left( u _ { n } \right)$? Justify your answer.
4. In 2020, an animal species numbered 10000 individuals. The evolution observed in previous years leads to the estimate that from 2021 onwards, this population will decrease by $5\%$ at the beginning of each year. To slow down this decline, it was decided to reintroduce 200 individuals at the end of each year, starting from 2021.
   A representative of an association supporting this strategy claims that: ``the species should not become extinct, but unfortunately, we will not prevent a loss of more than half the population''. What do you think of this statement? Justify your answer.

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.

In this exercise, we consider the sequence ( $T _ { n }$ ) defined by:
$$T _ { 0 } = 180 \mathrm { and } , \text { for all natural integer } n , T _ { n + 1 } = 0,955 T _ { n } + 0,9$$

1. a. Prove by induction that, for all natural integer $n , T _ { n } \geqslant 20$. b. Verify that for all natural integer $n , T _ { n + 1 } - T _ { n } = - 0,045 \left( T _ { n } - 20 \right)$. Deduce the direction of variation of the sequence ( $T _ { n }$ ). c. Conclude from the above that the sequence ( $T _ { n }$ ) is convergent. Justify.
   
2. For all natural integer $n$, we set: $u _ { n } = T _ { n } - 20$. a. Show that the sequence ( $u _ { n }$ ) is a geometric sequence and specify its common ratio. b. Deduce that for all natural integer $n , T _ { n } = 20 + 160 \times 0,955 ^ { n }$. c. Calculate the limit of the sequence ( $T _ { n }$ ). d. Solve the inequality $T _ { n } \leqslant 120$ with unknown $n$ a natural integer.
   
3. In this part, we are interested in the evolution of temperature at the center of a cake after it comes out of the oven. We consider that when the cake comes out of the oven, the temperature at the center of the cake is $180 ^ { \circ } \mathrm { C }$ and that of the ambient air is $20 ^ { \circ } \mathrm { C }$. Newton's law of cooling allows us to model the temperature at the center of the cake by the previous sequence ( $T _ { n }$ ). More precisely, $T _ { n }$ represents the temperature at the center of the cake, expressed in degrees Celsius, $n$ minutes after it comes out of the oven. a. Explain why the limit of the sequence ( $T _ { n }$ ) determined in question 2. c. was foreseeable in the context of the exercise. b. We consider the following Python function:

\begin{verbatim} def temp(x) : T = 180 n = 0 while T > x : T=0.955*T+0.9 n=n+1 return n \end{verbatim}
Give the result obtained by executing the command temp(120). Interpret the result in the context of the exercise.

Exercise 2 Sequences
Let $\left(u_{n}\right)$ be the sequence defined by $u_{0} = 4$ and, for every natural integer $n$, $u_{n+1} = \frac{1}{5} u_{n}^{2}$.

1. a. Calculate $u_{1}$ and $u_{2}$. b. Copy and complete the function below written in Python language. This function is named suite\_u and takes as parameter the natural integer $p$. It returns the value of the term of rank $p$ of the sequence $(u_{n})$. \begin{verbatim} def suite_u(p) : u= ... for i in range(1,...) : u =... return u \end{verbatim}
2. a. Prove by induction that for every natural integer $n$, $0 < u_{n} \leqslant 4$. b. Prove that the sequence $(u_{n})$ is decreasing. c. Deduce from this that the sequence $(u_{n})$ is convergent.
3. a. Justify that the limit $\ell$ of the sequence $(u_{n})$ satisfies the equality $\ell = \frac{1}{5} \ell^{2}$. b. Deduce from this the value of $\ell$.
4. For every natural integer $n$, we set $v_{n} = \ln\left(u_{n}\right)$ and $w_{n} = v_{n} - \ln(5)$. a. Show that, for every natural integer $n$, $v_{n+1} = 2v_{n} - \ln(5)$. b. Show that the sequence $(w_{n})$ is geometric with common ratio 2. c. For every natural integer $n$, give the expression of $w_{n}$ as a function of $n$ and show that $v_{n} = \ln\left(\frac{4}{5}\right) \times 2^{n} + \ln(5)$.
5. Calculate $\lim_{n \rightarrow +\infty} v_{n}$ and find again $\lim_{n \rightarrow +\infty} u_{n}$.

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 are given the sequence $( u _ { n } )$ defined by: $u _ { 0 } = 0$ and for every natural integer $n$, $u _ { n + 1 } = \frac { 1 } { 2 } u _ { n } + 1$. The sequence $\left( v _ { n } \right)$, defined for every natural integer $n$ by $v _ { n } = u _ { n } - 2$, is: a. arithmetic with common difference $- 2$; b. geometric with common ratio $- 2$; c. arithmetic with common difference $1$; d. geometric with common ratio $\frac { 1 } { 2 }$.

In a large French city, electric scooters are made available to users. A company, responsible for maintaining the scooter fleet, checks their condition every Monday.
It is estimated that:

• when a scooter is in good condition on a Monday, the probability that it is still in good condition the following Monday is 0.9;
• when a scooter is in poor condition on a Monday, the probability that it is in good condition the following Monday is 0.4.

We are interested in the condition of a scooter during the inspection phases. Let $n$ be a natural integer. We denote $B_n$ the event ``the scooter is in good condition $n$ weeks after its commissioning'' and $p_n$ the probability of $B_n$. When commissioned, the scooter is in good condition. We therefore have $p_0 = 1$.

1. Give $p_1$ and show that $p_2 = 0.85$. You may rely on a weighted tree.
2. Copy and complete the weighted tree.
3. Deduce that, for all natural integer $n$, $p_{n+1} = 0.5p_n + 0.4$.
4. a. Prove by induction that for all natural integer $n$, $p_n \geqslant 0.8$. b. Based on this result, what communication can the company consider to highlight the reliability of the fleet?
5. a. Consider the sequence $(u_n)$ defined for all natural integer $n$ by $u_n = p_n - 0.8$. Show that $(u_n)$ is a geometric sequence and give its first term and common ratio. b. Deduce the expression of $u_n$ then of $p_n$ as a function of $n$. c. Deduce the limit of the sequence $(p_n)$.

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?

Exercise 1 — 5 points Theme: probability, sequences
Parts A and B can be treated independently
Part A
Each day, an athlete must jump over a hurdle at the end of training. Based on the previous season, his coach estimates that

• if the athlete clears the hurdle one day, then he will clear it in $90\%$ of cases the next day;
• if the athlete does not clear the hurdle one day, then in $70\%$ of cases he will not clear it the next day either.

For every natural integer $n$, we denote:

• $R_{n}$ the event: ``The athlete successfully clears the hurdle during the $n$-th session'',
• $p_{n}$ the probability of event $R_{n}$. We consider that $p_{0} = 0.6$.

1. Let $n$ be a natural integer, copy the weighted tree below and complete the blanks.
2. Justify using the tree that, for every natural integer $n$, we have: $$p_{n+1} = 0.6 p_{n} + 0.3 .$$
3. Consider the sequence $(u_{n})$ defined, for every natural integer $n$, by $u_{n} = p_{n} - 0.75$. a. Prove that the sequence $(u_{n})$ is a geometric sequence and specify its common ratio and first term. b. Prove that, for every natural integer $n$: $$p_{n} = 0.75 - 0.15 \times 0.6^{n} .$$ c. Deduce that the sequence $(p_{n})$ is convergent and determine its limit $\ell$. d. Interpret the value of $\ell$ in the context of the exercise.

Part B
After many training sessions, the coach now estimates that the athlete clears each hurdle with a probability of 0.75 and this independently of whether or not he cleared the previous hurdles. We denote $X$ the random variable that gives the number of hurdles cleared by the athlete at the end of a 400 metres hurdles race which has 10 hurdles.

1. Specify the nature and parameters of the probability distribution followed by $X$.
2. Determine, to $10^{-3}$ near, the probability that the athlete clears all 10 hurdles.
3. Calculate $p(X \geqslant 9)$, to $10^{-3}$ near.

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.

Consider the sequence $(u_n)$ defined for every natural number $n$ by: $$u_n = \mathrm{e}^{2n+1}$$ The sequence $(u_n)$ is: a. arithmetic with common difference 2; b. geometric with common ratio e; c. geometric with common ratio $\mathrm{e}^2$; d. convergent to e.

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

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?