Exercise 4 — Candidates who have not followed the specialization course

For each of the following statements, indicate whether it is true or false, by justifying the answer. One point is awarded for each correct answer that is properly justified. An unjustified answer is not taken into account. An absence of an answer is not penalized.

1. Let $\left( u _ { n } \right)$ be the sequence defined by $$u _ { 0 } = 4 \text { and for all natural integer } n , u _ { n + 1 } = - \frac { 2 } { 3 } u _ { n } + 1$$ and let $( \nu _ { n } )$ be the sequence defined by $$\text { for all natural integer } n , v _ { n } = u _ { n } - \frac { 2 } { 3 }$$ Statement 1: The sequence $\left( v _ { n } \right)$ is a geometric sequence.
2. Let $( w _ { n } )$ be the sequence defined by, for all non-zero natural integer $n$, $$w _ { n } = \frac { 3 + \cos ( n ) } { n ^ { 2 } } .$$ Statement 2: The sequence $\left( w _ { n } \right)$ converges to 0.
3. Consider the following algorithm: $$\begin{aligned} & U \leftarrow 5 \\ & N \leftarrow 0 \end{aligned}$$ While $U \leqslant 5000$ $$\begin{aligned} & U \leftarrow 3 \times U - 8 \\ & N \leftarrow N + 1 \end{aligned}$$ End While Statement 3: At the end of execution, the variable $U$ contains the value 5000.
4. We denote $\mathbb { C }$ the set of complex numbers. We consider the equation (E) with unknown $z$ in $\mathbb { C }$ $$( z - \mathrm { i } ) \left( z ^ { 2 } + z \sqrt { 3 } + 1 \right) = 0$$ Statement 4: All solutions of equation (E) have modulus 1.
5. We consider the complex numbers $z _ { n }$ defined by $$z _ { 0 } = 2 \text { and for all natural integer } n , z _ { n + 1 } = 2 \mathrm { e } ^ { \mathrm { i } \frac { \pi } { 2 } } z _ { n } .$$ We equip the complex plane with a direct orthonormal coordinate system ( $\mathrm { O } ; \vec { u } , \vec { v }$ ). For all natural integer $n$, we denote $M _ { n }$ the point with affixe $z _ { n }$. Statement 5: For all natural integer $n$, the point O is the midpoint of the segment $\left[ M _ { n } M _ { n + 2 } \right]$.

We consider the sequence $\left( u _ { n } \right)$ defined, for all non-zero natural integers $n$, by:
$$u _ { n } = \frac { n ( n + 2 ) } { ( n + 1 ) ^ { 2 } }$$
The sequence $( v _ { n } )$ is defined by: $v _ { 1 } = u _ { 1 }$, $v _ { 2 } = u _ { 1 } \times u _ { 2 }$ and for all natural integers $n \geqslant 3$, $v _ { n } = u _ { 1 } \times u _ { 2 } \times \ldots \times u _ { n } = v _ { n - 1 } \times u _ { n }$.

1. Verify that we have $v _ { 2 } = \frac { 2 } { 3 }$ then calculate $v _ { 3 }$.
2. We consider the incomplete algorithm below. Copy and complete this algorithm on your paper so that, after its execution, the variable $V$ contains the value $v _ { n }$ where $n$ is a non-zero natural integer defined by the user. No justification is required.
   

   	Algorithm
   1.	$V \leftarrow 1$
   2.	For $i$ varying from 1 to $n$
   3.	$U \leftarrow \frac { \ldots ( \ldots + 2 ) } { ( \ldots + 1 ) ^ { 2 } }$
   4.	$V \leftarrow \ldots$
   5.	End For

   
   
3. a. Show that, for all non-zero natural integers $n$, $u _ { n } = 1 - \frac { 1 } { ( n + 1 ) ^ { 2 } }$. b. Show that, for all non-zero natural integers $n$, $0 < u _ { n } < 1$.
4. a. Show that the sequence $( v _ { n } )$ is decreasing. b. Justify that the sequence $( v _ { n } )$ is convergent (we do not ask to calculate its limit).
5. a. Verify that, for all non-zero natural integers $n$, $v _ { n + 1 } = v _ { n } \times \frac { ( n + 1 ) ( n + 3 ) } { ( n + 2 ) ^ { 2 } }$. b. Show by induction that, for all non-zero natural integers $n$, $v _ { n } = \frac { n + 2 } { 2 ( n + 1 ) }$. c. Determine the limit of the sequence $\left( v _ { n } \right)$.
6. We consider the sequence $w _ { n }$ defined by $w _ { 1 } = \ln \left( u _ { 1 } \right)$, $w _ { 2 } = \ln \left( u _ { 1 } \right) + \ln \left( u _ { 2 } \right)$ and, for all natural integers $n \geqslant 3$, by $$w _ { n } = \sum _ { k = 1 } ^ { n } \ln \left( u _ { k } \right) = \ln \left( u _ { 1 } \right) + \ln \left( u _ { 2 } \right) + \ldots + \ln \left( u _ { n } \right)$$ Show that $w _ { 7 } = 2 w _ { 1 }$.

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.

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: $$u_{n+1} = u_n + 0.05\left(20 - u_n\right) \text{ for every natural integer } n.$$

1. Verify that $u_1 = 1.95$.
2. 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 $n$, $u_n = 20 - 19 \times 0.95^n$.
3. Determine the limit of the sequence $(u_n)$.

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 $$L(t) = 20 - 19\mathrm{e}^{-0.05t}$$

1. Verify that the function $L$ is a solution of $(E)$ and that we also have $L(0) = 1$.
2. 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?

Let $f$ be the function defined on the interval $] - \frac { 1 } { 3 } ; + \infty [$ by: $$f ( x ) = \frac { 4 x } { 1 + 3 x }$$ We consider the sequence $(u _ { n })$ defined by: $u _ { 0 } = \frac { 1 } { 2 }$ and, for every natural number $n$, $u _ { n + 1 } = f \left( u _ { n } \right)$.

1. Calculate $u _ { 1 }$.
2. We admit that the function $f$ is increasing on the interval $] - \frac { 1 } { 3 } ; + \infty [$. a. Show by induction that, for every natural number $n$, we have: $\frac { 1 } { 2 } \leqslant u _ { n } \leqslant u _ { n + 1 } \leqslant 2$. b. Deduce that the sequence $(u _ { n })$ is convergent. c. We call $\ell$ the limit of the sequence $(u _ { n })$. Determine the value of $\ell$.
3. a. Copy and complete the Python function below which, for every positive real number $E$, determines the smallest value $P$ such that: $1 - u _ { P } < E$. \begin{verbatim} def seuil(E) : u=0.5 n = 0 while u = n = n + 1 return n \end{verbatim} b. Give the value returned by this program in the case where $E = 10 ^ { - 4 }$.
4. We consider the sequence $(v _ { n })$ defined, for every natural number $n$, by: $$v _ { n } = \frac { u _ { n } } { 1 - u _ { n } }$$ a. Show that the sequence $(v _ { n })$ is geometric with common ratio 4. Deduce, for every natural number $n$, the expression of $v _ { n }$ as a function of $n$. b. Prove that, for every natural number $n$, we have: $u _ { n } = \frac { v _ { n } } { v _ { n } + 1 }$. c. Show then that, for every natural number $n$, we have: $$u _ { n } = \frac { 1 } { 1 + 0.25 ^ { n } }$$ Find by calculation the limit of the sequence $(u _ { n })$.

Main topics covered: Numerical sequences; proof by induction.
We consider the sequences $(u_n)$ and $(v_n)$ defined by:
$$u_0 = 16 \quad; \quad v_0 = 5$$
and for any natural number $n$:
$$\left\{\begin{aligned} u_{n+1} & = \frac{3u_n + 2v_n}{5} \\ v_{n+1} & = \frac{u_n + v_n}{2} \end{aligned}\right.$$

1. Calculate $u_1$ and $v_1$.
2. We consider the sequence $(w_n)$ defined for any natural number $n$ by: $w_n = u_n - v_n$. a. Prove that the sequence $(w_n)$ is geometric with ratio 0.1.
   Deduce from this, for any natural number $n$, the expression of $w_n$ as a function of $n$. b. Specify the sign of the sequence $(w_n)$ and the limit of this sequence.
3. a. Prove that, for any natural number $n$, we have: $u_{n+1} - u_n = -0.4w_n$. b. Deduce that the sequence $(u_n)$ is decreasing.
   It can be shown in the same way that the sequence $(v_n)$ is increasing. We admit this result, and we note that we then have: for any natural number $n$, $v_n \geq v_0 = 5$. c. Prove by induction that, for any natural number $n$, we have: $u_n \geq 5$.
   Deduce that the sequence $(u_n)$ is convergent. We call $\ell$ the limit of $(u_n)$. It can be shown in the same way that the sequence $(v_n)$ is convergent. We admit this result, and we call $\ell'$ the limit of $(v_n)$.
4. a. Prove that $\ell = \ell'$. b. We consider the sequence $(c_n)$ defined for any natural number $n$ by: $c_n = 5u_n + 4v_n$. Prove that the sequence $(c_n)$ is constant, that is, for any natural number $n$, we have: $c_{n+1} = c_n$. Deduce that, for any natural number $n$, $c_n = 100$. c. Determine the common value of the limits $\ell$ and $\ell'$.

Main topics covered: Numerical sequences; proof by induction; geometric sequences.
The sequence $(u_{n})$ is defined on $\mathbb{N}$ by $u_{0} = 1$ and for every natural number $n$, $$u_{n+1} = \frac{3}{4}u_{n} + \frac{1}{4}n + 1.$$

1. Calculate, showing the calculations in detail, $u_{1}$ and $u_{2}$ in the form of irreducible fractions.

The extract, reproduced below, from a spreadsheet created with a spreadsheet application presents the values of the first terms of the sequence $(u_{n})$.

	A	B
1	$n$	$u_{n}$
2	0	1
3	1	1.75
4	2	2.5625
5	3	3.421875
6	4	4.31640625

1. a. What formula, then extended downward, can be written in cell B3 of the spreadsheet to obtain the successive terms of $(u_{n})$ in column B? b. Conjecture the direction of variation of the sequence $(u_{n})$.
2. a. Prove by induction that, for every natural number $n$, we have: $n \leqslant u_{n} \leqslant n+1$. b. Deduce from this, justifying the answer, the direction of variation and the limit of the sequence $(u_{n})$. c. Prove that: $$\lim_{n \rightarrow +\infty} \frac{u_{n}}{n} = 1$$
3. We denote by $(v_{n})$ the sequence defined on $\mathbb{N}$ by $v_{n} = u_{n} - n$ a. Prove that the sequence $(v_{n})$ is geometric with common ratio $\frac{3}{4}$. b. Deduce from this that, for every natural number $n$, we have: $u_{n} = \left(\frac{3}{4}\right)^{n} + n$.

Question 2: Consider the sequence $(v_n)$ defined on $\mathbb{N}$ by $v_n = \frac{3n}{n+2}$. We seek to determine the limit of $v_n$ as $n$ tends to $+\infty$.

a. $\lim_{n\rightarrow+\infty} v_n = 1$	b. $\lim_{n\rightarrow+\infty} v_n = 3$	c. $\lim_{n\rightarrow+\infty} v_n = \frac{3}{2}$	\begin{tabular}{l} d. We cannot
determine it

\hline \end{tabular}

Consider the sequence $(u_n)$ defined by: $u_0 = 1$ and, for every natural integer $n$, $$u_{n+1} = \frac{4u_n}{u_n + 4}.$$
1. The screenshot below presents the values, calculated using a spreadsheet, of the terms of the sequence $(u_n)$ for $n$ varying from 0 to 12, as well as those of the quotient $\frac{4}{u_n}$, (with, for the values of $u_n$, display of two digits for the decimal parts). Using these values, conjecture the expression of $\frac{4}{u_n}$ as a function of $n$. The purpose of this exercise is to prove this conjecture (question 5.) and to deduce the limit of the sequence $(u_n)$ (question 6.).

$n$	$u_n$	$\frac{4}{u_n}$
0	1,00	4
1	0,80	5
2	0,67	6
3	0,57	7
4	0,50	8
5	0,44	9
6	0,40	10
7	0,36	11
8	0,33	12
9	0,31	13
10	0,29	14
11	0,27	15
12	0,25	16

1. Prove by induction that, for every natural integer $n$, we have: $u_n > 0$.
2. Prove that the sequence $(u_n)$ is decreasing.
3. What can be concluded from questions 2. and 3. concerning the sequence $(u_n)$?
4. Consider the sequence $(v_n)$ defined for every natural integer $n$ by: $v_n = \frac{4}{u_n}$.

Prove that $(v_n)$ is an arithmetic sequence. Specify its common difference and its first term. Deduce, for every natural integer $n$, the expression of $v_n$ as a function of $n$.
6. Determine, for every natural integer $n$, the expression of $u_n$ as a function of $n$.
Deduce the limit of the sequence $(u_n)$.

We consider two sequences $(U _ { n })$ and $(V _ { n })$ defined on $\mathbb { N }$ such that:

• for every natural number $n$, $U _ { n } \leqslant V _ { n }$;
• $\lim _ { n \rightarrow + \infty } V _ { n } = 2$.

We can assert that: a. the sequence $(U _ { n })$ converges b. for every natural number $n$, $V _ { n } \leqslant 2$ c. the sequence $(U _ { n })$ diverges d. the sequence $(U _ { n })$ is bounded above

Exercise 1 (7 points) Theme: probabilities, sequences In a tourist region, a company offers a bicycle rental service for the day. The company has two distinct rental points, point A and point B. Bicycles can be borrowed and returned indifferently at either of the two rental points. It is assumed that the total number of bicycles is constant and that every morning, when the service opens, each bicycle is at point A or point B. According to a statistical study:

• If a bicycle is at point A one morning, the probability that it is at point A the next morning is equal to 0.84;
• If a bicycle is at point B one morning, the probability that it is at point B the next morning is equal to 0.76.

When the service opens on the first morning, the company has placed half of its bicycles at point A and the other half at point B. We consider a bicycle from the company chosen at random. For every positive integer $n$, the following events are defined:

• $A _ { n }$ : ``the bicycle is at point A on the $n$-th morning''
• $B _ { n }$ : ``the bicycle is at point B on the $n$-th morning''.

For every positive integer $n$, we denote by $a _ { n }$ the probability of event $A _ { n }$ and by $b _ { n }$ the probability of event $B _ { n }$. Thus $a _ { 1 } = 0.5$ and $b _ { 1 } = 0.5$.

1. Copy and complete the weighted tree that models the situation for the first two mornings.
2. a. Calculate $a _ { 2 }$. b. The bicycle is at point A on the second morning. Calculate the probability that it was at point B on the first morning. The probability will be rounded to the nearest thousandth.
3. a. Copy and complete the weighted tree that models the situation for the $n$-th and $(n + 1)$-th mornings. b. Justify that for every positive integer $n$, $a _ { n + 1 } = 0.6 a _ { n } + 0.24$.
4. Show by induction that, for every positive integer $n$, $a _ { n } = 0.6 - 0.1 \times 0.6 ^ { n - 1 }$.
5. Determine the limit of the sequence $(a _ { n })$ and interpret this limit in the context of the exercise.
6. Determine the smallest positive integer $n$ such that $a _ { n } \geqslant 0.599$ and interpret the result obtained in the context of the exercise.

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.

Exercise 2 — 7 points

Topics: Sequences, Functions This exercise is a multiple choice questionnaire. For each of the following questions, only one of the four proposed answers is correct.
A wrong answer, multiple answers, or no answer to a question earns neither points nor deducts points. To answer, indicate on your answer sheet the question number and the letter of the chosen answer. No justification is required.

1. We consider the sequences $\left( a _ { n } \right)$ and $\left( b _ { n } \right)$ defined by $a _ { 0 } = 1$ and, for every natural number $n$, $a _ { n + 1 } = 0.5 a _ { n } + 1$ and $b _ { n } = a _ { n } - 2$. We can affirm that: a. $\left( a _ { n } \right)$ is arithmetic; b. $\left( b _ { n } \right)$ is geometric; c. $\left( a _ { n } \right)$ is geometric; d. $\left( b _ { n } \right)$ is arithmetic.
2. In questions 2. and 3., we consider the sequences $\left( u _ { n } \right)$ and $\left( v _ { n } \right)$ defined by: $$u _ { 0 } = 2 , v _ { 0 } = 1 \mathrm { and } , \text { for every natural number } n : \left\{ \begin{array} { l } u _ { n + 1 } = u _ { n } + 3 v _ { n } \\ v _ { n + 1 } = u _ { n } + v _ { n } . \end{array} \right.$$ We can affirm that: a. $\left\{ \begin{array} { l } u _ { 2 } = 5 \\ v _ { 2 } = 3 \end{array} \right.$ b. $u _ { 2 } ^ { 2 } - 3 v _ { 2 } ^ { 2 } = - 2 ^ { 2 }$ c. $\frac { u _ { 2 } } { v _ { 2 } } = 1.75$ d. $5 u _ { 1 } = 3 v _ { 1 }$.
3. We consider the sequences $\left( u _ { n } \right)$ and $\left( v _ { n } \right)$ defined by: $$u _ { 0 } = 2 , v _ { 0 } = 1 \mathrm { and } , \text { for every natural number } n : \left\{ \begin{array} { l } u _ { n + 1 } = u _ { n } + 3 v _ { n } \\ v _ { n + 1 } = u _ { n } + v _ { n } . \end{array} \right.$$ We consider the program below written in Python language: \begin{verbatim} def valeurs() : u = 2 v = 1 for k in range(1,11) c = u u = u + 3*v v = c + v return (u, v) \end{verbatim} This program returns: a. $u _ { 11 }$ and $v _ { 11 }$; b. $u _ { 10 }$ and $v _ { 11 }$; c. the values of $u _ { n }$ and $v _ { n }$ for $n$ ranging from 1 to 10; d. $u _ { 10 }$ and $v _ { 10 }$.
4. For questions 4. and 5., we consider a function $f$ twice differentiable on the interval $[-4; 2]$. We denote by $f ^ { \prime }$ the derivative function of $f$ and $f ^ { \prime \prime }$ the second derivative of $f$. We are given below the representative curve $\mathscr { C } ^ { \prime }$ of the derivative function $f ^ { \prime }$ in a coordinate system of the plane. We are also given the points $\mathrm { A } ( - 2 ; 0 ) , \mathrm { B } ( 1 ; 0 )$ and $\mathrm { C } ( 0 ; 5 )$. The function $f$ is: a. concave on $[-2; 1]$; b. convex on $[-4; 0]$; c. convex on $[ - 2 ; 1 ]$; d. convex on $[ 0 ; 2 ]$.
5. For questions 4. and 5., we consider a function $f$ twice differentiable on the interval $[-4; 2]$. We denote by $f ^ { \prime }$ the derivative function of $f$ and $f ^ { \prime \prime }$ the second derivative of $f$. We are given below the representative curve $\mathscr { C } ^ { \prime }$ of the derivative function $f ^ { \prime }$ in a coordinate system of the plane. We are also given the points $\mathrm { A } ( - 2 ; 0 ) , \mathrm { B } ( 1 ; 0 )$ and $\mathrm { C } ( 0 ; 5 )$. We admit that the line (BC) is tangent to the curve $\mathscr { C } ^ { \prime }$ at point B. We have: a. $f ^ { \prime } ( 1 ) < 0$; b. $f ^ { \prime } ( 1 ) = 5$; c. $f ^ { \prime \prime } ( 1 ) > 0$; d. $f ^ { \prime \prime } ( 1 ) = - 5$.
6. Let $f$ be the function defined on $\mathbb { R }$ by $f ( x ) = \left( x ^ { 2 } + 1 \right) \mathrm { e } ^ { x }$. The antiderivative $F$ of $f$ on $\mathbb { R }$ such that $F ( 0 ) = 1$ is defined by: a. $F ( x ) = \left( x ^ { 2 } - 2 x + 3 \right) \mathrm { e } ^ { x }$; b. $F ( x ) = \left( x ^ { 2 } - 2 x + 3 \right) \mathrm { e } ^ { x } - 2$; c. $F ( x ) = \left( \frac { 1 } { 3 } x ^ { 3 } + x \right) \mathrm { e } ^ { x } + 1$; d. $F ( x ) = \left( \frac { 1 } { 3 } x ^ { 3 } + x \right) \mathrm { e } ^ { x }$.

Exercise 2 (7 points) Themes: numerical functions and sequences This exercise is a multiple choice questionnaire. For each of the following questions, only one of the four proposed answers is correct. A wrong answer, multiple answers, or no answer to a question earns neither points nor deducts points. To answer, indicate on your paper the question number and the letter of the chosen answer. No justification is required.
For questions 1 to 3 below, consider a function $f$ defined and twice differentiable on $\mathbb{R}$. The curve of its derivative function $f'$ is given. We admit that $f'$ has a maximum at $-\frac{3}{2}$ and that its curve intersects the x-axis at the point with coordinates $\left(-\frac{1}{2}; 0\right)$.
Question 1: a. The function $f$ has a maximum at $-\frac{3}{2}$; b. The function $f$ has a maximum at $-\frac{1}{2}$; c. The function $f$ has a minimum at $-\frac{1}{2}$; d. At the point with abscissa $-1$, the curve of function $f$ has a horizontal tangent.
Question 2: a. The function $f$ is convex on $]-\infty; -\frac{3}{2}[$; c. The curve $\mathscr{C}_f$ representing function $f$ does not have an inflection point;
Question 3: The second derivative $f''$ of function $f$ satisfies: a. $f''(x) \geqslant 0$ for $x \in ]-\infty; -\frac{1}{2}[$; b. $f''(x) \geqslant 0$ for $x \in [-2; -1]$; c. $f''\left(-\frac{3}{2}\right) = 0$; d. $f''(-3) = 0$.
Question 4: Consider three sequences $\left(u_n\right)$, $\left(v_n\right)$ and $\left(w_n\right)$. We know that, for every natural number $n$, we have: $u_n \leqslant v_n \leqslant w_n$ and furthermore: $\lim_{n \rightarrow +\infty} u_n = 1$ and $\lim_{n \rightarrow +\infty} w_n = 3$. We can then affirm that: a. the sequence $\left(v_n\right)$ converges; b. If the sequence $(u_n)$ is increasing then the sequence $(v_n)$ is bounded below by $u_0$; c. $1 \leqslant v_0 \leqslant 3$; d. the sequence $(v_n)$ diverges.
Question 5: Consider a sequence $(u_n)$ such that, for every non-zero natural number $n$: $u_n \leqslant u_{n+1} \leqslant \frac{1}{n}$. We can then affirm that: a. the sequence $(u_n)$ diverges; b. the sequence $(u_n)$ converges; c. $\lim_{n \rightarrow +\infty} u_n = 0$; d. $\lim_{n \rightarrow +\infty} u_n = 1$.
Question 6: Consider $(u_n)$ a real sequence such that for every natural number $n$, we have: $n < u_n < n+1$. We can affirm that: a. There exists a natural number $N$ such that $u_N$ is an integer; b. the sequence $(u_n)$ is increasing; c. the sequence $(u_n)$ is convergent; d. The sequence $(u_n)$ has no limit.

Exercise 2 (7 points) -- Sequences, functions
Let $k$ be a real number. Consider the sequence $\left(u_n\right)$ defined by its first term $u_0$ and for every natural number $n$, $$u_{n+1} = k u_n \left(1 - u_n\right)$$
The two parts of this exercise are independent. We study two cases depending on the values of $k$.
Part 1
In this part, $k = 1.9$ and $u_0 = 0.1$. Therefore, for every natural number $n$, $u_{n+1} = 1.9 u_n \left(1 - u_n\right)$.

1. Consider the function $f$ defined on $[0; 1]$ by $f(x) = 1.9x(1 - x)$. a. Study the variations of $f$ on the interval $[0; 1]$. b. Deduce that if $x \in [0; 1]$ then $f(x) \in [0; 1]$.
2. Below are represented the first terms of the sequence $\left(u_n\right)$ constructed from the curve $\mathscr{C}_f$ of the function $f$ and the line $D$ with equation $y = x$. Conjecture the direction of variation of the sequence $(u_n)$ and its possible limit.
3. a. Using the results from question 1, prove by induction that for every natural number $n$: $$0 \leqslant u_n \leqslant u_{n+1} \leqslant \frac{1}{2}$$ b. Deduce that the sequence $(u_n)$ converges. c. Determine its limit.

Part 2
In this part, $k = \frac{1}{2}$ and $u_0 = \frac{1}{4}$. Therefore, for every natural number $n$, $u_{n+1} = \frac{1}{2} u_n \left(1 - u_n\right)$ and $u_0 = \frac{1}{4}$. We admit that for every natural number $n$: $0 \leqslant u_n \leqslant \left(\frac{1}{2}\right)^n$.

1. Prove that the sequence $(u_n)$ converges and determine its limit.
2. Consider the Python function \texttt{algo(p)} where \texttt{p} denotes a non-zero natural number: \begin{verbatim} def algo(p) : u = 1/4 n = 0 while u > 10**(-p): u = 1/2*u*(1 - u) n = n+1 return(n) \end{verbatim} Explain why, for every non-zero natural number $p$, the while loop does not run indefinitely, which allows the command \texttt{algo(p)} to return a value.

Exercise 3: Sequences
The population of an endangered species is closely monitored in a nature reserve. Climate conditions as well as poaching cause this population to decrease by $10\%$ each year. To compensate for these losses, 100 individuals are reintroduced into the reserve at the end of each year. We wish to study the evolution of the population size of this species over time. For this, we model the population size of the species by the sequence $(u_n)$ where $u_n$ represents the population size at the beginning of the year $2020 + n$. We admit that for all natural integer $n$, $u_n \geqslant 0$. At the beginning of the year 2020, the studied population has 2000 individuals, thus $u_0 = 2000$.

1. Justify that the sequence $(u_n)$ satisfies the recurrence relation: $$u_{n+1} = 0.9u_n + 100.$$
2. Calculate $u_1$ then $u_2$.
3. Prove by induction that for all natural integer $n$: $1000 < u_{n+1} \leqslant u_n$.
4. Is the sequence $(u_n)$ convergent? Justify your answer.
5. We consider the sequence $(v_n)$ defined for all natural integer $n$ by $v_n = u_n - 1000$. a. Show that the sequence $(v_n)$ is geometric with common ratio 0.9. b. Deduce that, for all natural integer $n$, $u_n = 1000\left(1 + 0.9^n\right)$. c. Determine the limit of the sequence $(u_n)$. Give an interpretation of this in the context of this exercise.
6. We wish to determine the number of years necessary for the population size to fall below a certain threshold $S$ (with $S > 1000$). a. Determine the smallest integer $n$ such that $u_n \leqslant 1020$. Justify your answer by a calculation. b. In the Python program opposite, the variable $n$ denotes the number of years elapsed since 2020, the variable $u$ denotes the population size. Copy and complete this program so that it returns the number of years necessary for the population size to fall below the threshold $S$. \begin{verbatim} def population(S) : n=0 u=2000 while ......: u= ... n = ... return ... \end{verbatim}

Let $\left(u_n\right)$ be the sequence defined by $u_0 = 1$ and for every natural integer $n$ $$u_{n+1} = \frac{u_n}{1 + u_n}$$

1. a. Calculate the terms $u_1, u_2$ and $u_3$. Give the results as irreducible fractions. b. Copy the Python script below and complete lines 3 and 6 so that \texttt{liste($k$)} takes as parameter a natural integer $k$ and returns the list of the first values of the sequence $\left(u_n\right)$ from $u_0$ to $u_k$. \begin{verbatim} def liste(k) : L = [] u=... for i in range(0, k+1) : L.append(u) u = ... return(L) \end{verbatim}
   
2. It is admitted that, for every natural integer $n$, $u_n$ is strictly positive. Determine the direction of variation of the sequence $(u_n)$.
3. Deduce that the sequence $(u_n)$ converges.
4. Determine the value of its limit.
5. a. Conjecture an expression of $u_n$ as a function of $n$. b. Prove by induction the previous conjecture.

At the beginning of 2021, a bird colony had 40 individuals. Observation leads to modeling the population evolution by the sequence $(u_n)$ defined for all natural integers $n$ by: $$\begin{cases} u _ { 0 } & = 40 \\ u _ { n + 1 } & = 0,008 u _ { n } \left( 200 - u _ { n } \right) \end{cases}$$ where $u _ { n }$ denotes the number of individuals at the beginning of the year $( 2021 + n )$.

1. Give an estimate, according to this model, of the number of birds in the colony at the beginning of 2022.

Consider the function $f$ defined on the interval $[ 0 ; 100 ]$ by $f ( x ) = 0,008 x ( 200 - x )$.

1. Solve in the interval $[ 0 ; 100 ]$ the equation $f ( x ) = x$.
2. a. Prove that the function $f$ is increasing on the interval $[ 0 ; 100 ]$ and draw its variation table. b. By noting that, for all natural integers $n , u _ { n + 1 } = f \left( u _ { n } \right)$, prove by induction that, for all natural integers $n$: $$0 \leqslant u _ { n } \leqslant u _ { n + 1 } \leqslant 100$$ c. Deduce that the sequence $(u_n)$ is convergent. d. Determine the limit $\ell$ of the sequence $(u_n)$. Interpret the result in the context of the exercise.
3. Consider the following algorithm: \begin{verbatim} def seuil(p) : n=0 u=40 while u < p: n =n+1 u=0.008*u*(200-u) return(n+2021) \end{verbatim} The execution of seuil(100) returns no value. Explain why using question 3.

Exercise 4 (7 points) — Main topics covered: sequences, functions, antiderivatives.
This exercise is a multiple choice questionnaire. For each of the following questions, only one of the four proposed answers is correct. A wrong answer, multiple answers or the absence of an answer to a question neither awards nor deducts points. To answer, indicate on your paper the question number and the letter of the chosen answer. No justification is required.

1. Consider the sequence $(u_n)$ defined for all natural number $n$ by $$u_n = \frac{(-1)^n}{n+1}.$$ We can affirm that: a. the sequence $(u_n)$ diverges to $+\infty$. b. the sequence $(u_n)$ diverges to $-\infty$. c. the sequence $(u_n)$ has no limit. d. the sequence $(u_n)$ converges.
   In questions 2 and 3, we consider two sequences $(v_n)$ and $(w_n)$ satisfying the relation: $$w_n = \mathrm{e}^{-2v_n} + 2.$$
   
2. Let $a$ be a strictly positive real number. We have $v_0 = \ln(a)$. a. $w_0 = \dfrac{1}{a^2} + 2$ b. $w_0 = \dfrac{1}{a^2 + 2}$ c. $w_0 = -2a + 2$ d. $w_0 = \dfrac{1}{-2a} + 2$
   
3. We know that the sequence $(v_n)$ is increasing. We can affirm that the sequence $(w_n)$ is: a. decreasing and bounded above by 3. b. decreasing and bounded below by 2. c. increasing and bounded above by 3. d. increasing and bounded below by 2.
   
4. Consider the sequence $(a_n)$ defined as follows: $$a_0 = 2 \text{ and, for all natural number } n, \quad a_{n+1} = \frac{1}{3}a_n + \frac{8}{3}.$$ For all natural number $n$, we have: a. $a_n = 4 \times \left(\dfrac{1}{3}\right)^n - 2$ b. $a_n = -\dfrac{2}{3^n} + 4$ c. $a_n = 4 - \left(\dfrac{1}{3}\right)^n$ d. $a_n = 2 \times \left(\dfrac{1}{3}\right)^n + \dfrac{8n}{3}$
   
5. Consider a sequence $(b_n)$ such that, for all natural number $n$, we have: $$b_{n+1} = b_n + \ln\left(\frac{2}{(b_n)^2 + 3}\right)$$ We can affirm that: a. the sequence $(b_n)$ is increasing. b. the sequence $(b_n)$ is decreasing. c. the sequence $(b_n)$ is not monotone. d. the direction of variation of the sequence $(b_n)$ depends on $b_0$.
   
6. Consider the function $g$ defined on the interval $]0; +\infty[$ by: $$g(x) = \frac{\mathrm{e}^x}{x}$$ We denote $\mathcal{C}_g$ the representative curve of the function $g$ in an orthogonal coordinate system. The curve $\mathcal{C}_g$ has: a. a vertical asymptote and a horizontal asymptote. b. a vertical asymptote and no horizontal asymptote. c. no vertical asymptote and a horizontal asymptote. d. no vertical asymptote and no horizontal asymptote.
   
7. Let $f$ be the function defined on $\mathbb{R}$ by $$f(x) = x\mathrm{e}^{x^2+1}$$ Let $F$ be an antiderivative on $\mathbb{R}$ of the function $f$. For all real $x$, we have: a. $F(x) = \dfrac{1}{2}x^2\mathrm{e}^{x^2+1}$ b. $F(x) = \left(1 + 2x^2\right)\mathrm{e}^{x^2+1}$ c. $F(x) = \mathrm{e}^{x^2+1}$ d. $F(x) = \dfrac{1}{2}\mathrm{e}^{x^2+1}$

Consider a sequence $( u _ { n } )$ such that, for every natural integer, we have: $$1 + \left( \frac { 1 } { 4 } \right) ^ { n } \leqslant u _ { n } \leqslant 2 - \frac { n } { n + 1 }$$ We can affirm that the sequence $\left( u _ { n } \right)$: a. converges to $2$; b. converges to $1$; c. diverges to $+ \infty$; d. has no limit.

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?

Consider the numerical sequence $(u_n)$ defined for all natural integer $n$ by
$$u_n = \frac{1 + 2^n}{3 + 5^n}$$
This sequence: a. diverges to $+\infty$ b. converges to $\frac{2}{5}$ c. converges to 0 d. converges to $\frac{1}{3}$.

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.

We consider the sequence $(u_n)$ defined by $u_0 = 3$ and, for every natural integer $n$, by:
$$u_{n+1} = 5u_n - 4n - 3$$

1. a. Prove that $u_1 = 12$. b. Determine $u_2$ by detailing the calculation. c. Using a calculator, conjecture the direction of variation and the limit of the sequence $(u_n)$.
2. a. Prove by induction that, for every natural integer $n$, we have: $$u_n \geqslant n + 1.$$ b. Deduce the limit of the sequence $(u_n)$.
3. We consider the sequence $(v_n)$ defined for every natural integer $n$ by: $$v_n = u_n - n - 1$$ a. Prove that the sequence $(v_n)$ is geometric. Give its common ratio and its first term $v_0$. b. Deduce, for every natural integer $n$, the expression of $v_n$ as a function of $n$. c. Deduce that for every natural integer $n$: $$u_n = 2 \times 5^n + n + 1$$ d. Deduce the direction of variation of the sequence $(u_n)$.
4. We consider the function below, written incompletely in Python language and intended to return the smallest natural integer $n$ such that $u_n \geqslant 10^7$. a. Copy the program and complete the two missing instructions. b. What is the value returned by this function? \begin{verbatim} def suite() : u = 3 n = 0 while...: u = ... n = n + 1 return n \end{verbatim}