Exercise 5 — Candidates who have not chosen the specialisation option
Let ( $v _ { n }$ ) be the sequence defined by
$$v _ { 1 } = \ln ( 2 ) \text { and, for all natural integer } n \text { non-zero, } v _ { n + 1 } = \ln \left( 2 - \mathrm { e } ^ { - v _ { n } } \right) .$$
We admit that this sequence is defined for all non-zero natural integer $n$. We then define the sequence ( $S _ { n }$ ) for all non-zero natural integer $n$ by :
$$S _ { n } = \sum _ { k = 1 } ^ { n } v _ { k } = v _ { 1 } + v _ { 2 } + \cdots + v _ { n } .$$
The purpose of this exercise is to determine the limit of ( $S _ { n }$ ).

Part A - Conjectures using an algorithm

1. Copy and complete the following algorithm which calculates and displays the value of $S _ { n }$ for a value of $n$ chosen by the user :
   

   Variables :

   $n , k$ integers

   $S , v$ real numbers

   Initialisation :

   Input the value of $n$

   $v$ takes the value $\ldots$

   $S$ takes the value $\ldots$

   Processing:

   For $k$ varying from \ldots to \ldots do

   \ldots takes the value \ldots

   \ldots takes the value \ldots

   End For

   Output :

   Display $S$

   
   
2. Using this algorithm, we obtain some values of $S _ { n }$. The values rounded to the nearest tenth are given in the table below :
   

   $n$	10	100	1000	10000	100000	1000000
   $S _ { n }$	2.4	4.6	6.9	9.2	11.5	13.8

   
   By explaining your approach, make a conjecture about the behaviour of the sequence $\left( S _ { n } \right)$.

Part B - Study of an auxiliary sequence

For all non-zero natural integer $n$, we define the sequence ( $u _ { n }$ ) by $u _ { n } = \mathrm { e } ^ { v _ { n } }$.

1. Verify that $u _ { 1 } = 2$ and that, for all non-zero natural integer $n$, $u _ { n + 1 } = 2 - \frac { 1 } { u _ { n } }$.
2. Calculate $u _ { 2 } , u _ { 3 }$ and $u _ { 4 }$. Results should be given in fractional form.
3. Prove that, for all non-zero natural integer $n$, $u _ { n } = \frac { n + 1 } { n }$.

Part C - Study of ( $S _ { n }$ )

1. For all non-zero natural integer $n$, express $v _ { n }$ as a function of $u _ { n }$, then $v _ { n }$ as a function of $n$.
2. Verify that $S _ { 3 } = \ln ( 4 )$.
3. For all non-zero natural integer $n$, express $S _ { n }$ as a function of $n$. Deduce the limit of the sequence $\left( S _ { n } \right)$.

The sequence ( $u _ { n }$ ) is defined by:
$$u _ { 0 } = 0 \quad \text { and, for all natural integer } n , u _ { n + 1 } = \frac { 1 } { 2 - u _ { n } } .$$

1. a. Using the calculation of the first terms of the sequence ( $u _ { n }$ ), conjecture the explicit form of $u _ { n }$ as a function of $n$. Prove this conjecture. b. Deduce the value of the limit $\ell$ of the sequence $\left( u _ { n } \right)$.
2. Complete, in appendix 2, the algorithm to determine the value of the smallest integer $n$ such that $\left| u _ { n + 1 } - u _ { n } \right| \leqslant 10 ^ { - 3 }$.

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

1. Show by induction that, for every natural integer $n$, $u_{n}$ belongs to $[0;1]$.
2. Study the variations of the sequence $(u_{n})$.
3. Show that the sequence $(u_{n})$ is convergent.
4. We denote by $\ell$ its limit, and we admit that $\ell$ satisfies the equality $f(\ell) = \ell$. Deduce the value of $\ell$.

(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.

(Candidates who have followed the specialization course)
We observe the size of an ant colony every day. For any non-zero natural number $n$, we denote by $u _ { n }$ the number of ants, expressed in thousands, in this population at the end of the $n$-th day. At the beginning of the study the colony has 5000 ants and after one day it has 5100 ants. Thus, we have $u _ { 0 } = 5$ and $u _ { 1 } = 5.1$. We assume that the increase in the size of the colony from one day to the next decreases by $10\%$ each day. In other words, for any natural number $n$,
$$u _ { n + 2 } - u _ { n + 1 } = 0.9 \left( u _ { n + 1 } - u _ { n } \right) .$$

1. Prove that under these conditions, $u _ { 2 } = 5.19$.
2. For any natural number $n$, we set $V _ { n } = \binom { u _ { n + 1 } } { u _ { n } }$ and $A = \left( \begin{array} { c c } 1.9 & - 0.9 \\ 1 & 0 \end{array} \right)$. a. Prove that, for any natural number $n$, we have $V _ { n + 1 } = A V _ { n }$.

We then admit that, for any natural number $n$, $V _ { n } = A ^ { n } V _ { 0 }$. b. We set $P = \left( \begin{array} { c c } 0.9 & 1 \\ 1 & 1 \end{array} \right)$. We admit that the matrix $P$ is invertible.
Using a calculator, determine the matrix $P ^ { - 1 }$. By detailing the calculations, determine the matrix $D$ defined by $D = P ^ { - 1 } A P$. c. Prove by induction that, for any natural number $n$, we have $A ^ { n } = P D ^ { n } P ^ { - 1 }$. For any natural number $n$, we admit that
$$A ^ { n } = \left( \begin{array} { c c } - 10 \times 0.9 ^ { n + 1 } + 10 & 10 \times 0.9 ^ { n + 1 } - 9 \\ - 10 \times 0.9 ^ { n } + 10 & 10 \times 0.9 ^ { n } - 9 \end{array} \right) .$$
d. Deduce that, for any natural number $n$, $u _ { n } = 6 - 0.9 ^ { n }$.

1. Calculate the size of the colony at the end of the $10 ^ { \mathrm { th } }$ day. Round the result to the nearest ant.
2. Calculate the limit of the sequence $(u _ { n })$. Interpret this result in context.

The plane is equipped with an orthonormal coordinate system ( $\mathrm { O } , \vec { u } , \vec { v }$ ). For all integer $n \geqslant 4$, we consider $P _ { n }$ a regular polygon with $n$ sides, with centre $O$ and whose area is equal to 1. We admit that such a polygon is made up of $n$ triangles superimposable to a given triangle $\mathrm { OA } _ { n } \mathrm {~B} _ { n }$, isosceles at O. We denote $r _ { n } = \mathrm { OA } _ { n }$ the distance between the centre O and the vertex $\mathrm { A } _ { n }$ of such a polygon.
Part A: study of the particular case $n = 6$

1. Justify the fact that the triangle $\mathrm { OA } _ { 6 } \mathrm {~B} _ { 6 }$ is equilateral, and that its area is equal to $\frac { 1 } { 6 }$.
2. Express as a function of $r _ { 6 }$ the height of the triangle $\mathrm { OA } _ { 6 } \mathrm {~B} _ { 6 }$ from the vertex $\mathrm { B } _ { 6 }$.
3. Deduce that $r _ { 6 } = \sqrt { \frac { 2 } { 3 \sqrt { 3 } } }$.

Part B: general case with $n \geqslant 4$
In the method considered, we take as initial matrix the matrix $I = \left( \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right)$.

1. Determine the two missing matrices $A$ and $B$, in the third row of the Stern-Brocot tree.
2. We associate to a matrix $M = \left( \begin{array} { l l } a & c \\ b & d \end{array} \right)$ of the Stern-Brocot tree the fraction $\frac { a + c } { b + d }$. Show that, in this association, the path ``left-right-left'' starting from the initial matrix in the tree, leads to a matrix corresponding to the fraction $\frac { 3 } { 5 }$.
3. Let $M = \left( \begin{array} { l l } a & c \\ b & d \end{array} \right)$ be a matrix of the tree. We recall that $a , b , c , d$ are integers. We denote $\Delta _ { M } = a d - b c$, the difference of the diagonal products of this matrix. a. Show that if $a d - b c = 1$, then $d ( a + c ) - c ( b + d ) = 1$. b. Deduce that if $M = \left( \begin{array} { l l } a & c \\ b & d \end{array} \right)$ is a matrix of the Stern-Brocot tree such that $\Delta _ { M } = a d - b c = 1$, then $\Delta _ { M \times G } = 1$, that is, the difference of the diagonal products of the matrix $M \times G$ is also equal to 1. We similarly admit that $\Delta _ { M \times D } = 1$, and that all other matrices $N$ of the Stern-Brocot tree satisfy the equality $\Delta _ { N } = 1$.
4. Deduce from the previous question that every fraction associated with a matrix of the Stern-Brocot tree is in lowest terms.
5. Let $m$ and $n$ be two non-zero natural integers that are coprime. Thus the fraction $\frac { m } { n }$ is in lowest terms. We consider the following algorithm: \begin{verbatim} VARIABLES : m and n are non-zero natural integers and coprime PROCESSING : While m = do If m

   Display		$\ldots$	$\ldots$	$\ldots$	$\ldots$
   $m$	4	$\ldots$	$\ldots$	$\ldots$	$\ldots$
   $n$	7	$\ldots$	$\ldots$	$\ldots$	$\ldots$

   
   b. Conjecture the role of this algorithm. Verify by a matrix calculation the result provided with the values $m = 4$ and $n = 7$.

Exercise 1 -- Part A

Consider the sequence $\left( u _ { n } \right)$ defined for every natural integer $n$ by: $$u _ { n } = \int _ { 0 } ^ { n } \mathrm { e } ^ { - x ^ { 2 } } \mathrm {~d} x$$ We will not attempt to calculate $u _ { n }$ as a function of $n$.

1. a. Show that the sequence $(u_n)$ is increasing. b. Prove that for every real number $x \geqslant 0$, we have: $- x ^ { 2 } \leqslant - 2 x + 1$, then: $$\mathrm { e } ^ { - x ^ { 2 } } \leqslant \mathrm { e } ^ { - 2 x + 1 }$$ Deduce that for every natural integer $n$, we have: $u _ { n } < \frac { \mathrm { e } } { 2 }$. c. Prove that the sequence $(u_n)$ is convergent. We will not attempt to calculate its limit.
   
2. In this question, we propose to obtain an approximate value of $u _ { 2 }$.
   In the orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath})$ below, we have drawn the curve $\mathscr{C}_f$ representing the function $f$ defined on the interval $[0;2]$ by $f(x) = \mathrm{e}^{-x^2}$, and the rectangle OABC where $\mathrm{A}(2;0)$, $\mathrm{B}(2;1)$ and $\mathrm{C}(0;1)$. We have shaded the region $\mathscr{D}$ between the curve $\mathscr{C}_f$, the horizontal axis, the vertical axis and the line with equation $x = 2$.
   Consider the random experiment consisting of choosing a point $M$ at random inside the rectangle OABC. We admit that the probability $p$ that this point belongs to the region is: $p = \frac{\text{area of } \mathscr{D}}{\text{area of } \mathrm{OABC}}$. a. Justify that $u_2 = 2p$. b. Consider the following algorithm:
   

   L1	Variables: $N, C$ integers; $X, Y, F$ real numbers
   L2	Input: Enter $N$
   L3	Initialization: $C$ takes the value 0
   L4	Processing:
   L5	For $k$ varying from 1 to $N$
   L6	$X$ takes the value of a random number between 0 and 2
   L7	$Y$ takes the value of a random number between 0 and 1
   L8	If $Y \leqslant \mathrm{e}^{-X^2}$ then
   L9	$C$ takes the value $C + 1$
   L10	End if
   L11	End for
   L12	Display $C$
   L13	$F$ takes the value $C/N$
   L14	Display $F$

   
   i. What does the condition on line $L8$ allow us to test regarding the position of point $M(X;Y)$? ii. Interpret the value $F$ displayed by this algorithm. iii. What can we conjecture about the value of $F$ when $N$ becomes very large? c. By running this algorithm for $N = 10^6$, we obtain $C = 441138$.
   We admit in this case that the value $F$ displayed by the algorithm is an approximate value of the probability $p$ to within $10^{-3}$. Deduce an approximate value of $u_2$ to within $10^{-2}$.

Part B

The sign, modeled by the region $\mathscr{D}$ defined in Part A, is cut from a rectangular sheet of 2 meters by 1 meter. It is represented in an orthonormal coordinate system $(\mathrm{O}, \vec{\imath}, \vec{\jmath})$; the chosen unit is the meter.
For $x$ a real number belonging to the interval $[0;2]$, we denote:

• $M$ the point on the curve $\mathscr{C}_f$ with coordinates $(x; \mathrm{e}^{-x^2})$,
• $N$ the point with coordinates $(x; 0)$,
• $P$ the point with coordinates $\left(0; \mathrm{e}^{-x^2}\right)$,
• $A(x)$ the area of rectangle $ONMP$.

1. Justify that for every real number $x$ in the interval $[0;2]$, we have: $A(x) = x\mathrm{e}^{-x^2}$.
2. Determine the position of point $M$ on the curve $\mathscr{C}_f$ for which the area of rectangle $ONMP$ is maximum.
3. The rectangle $ONMP$ of maximum area obtained in question 2. must be painted blue, and the rest of the sign in white. Determine, in $\mathrm{m}^2$ and to within $10^{-2}$, the measure of the surface to be painted blue and that of the surface to be painted white.

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 $(u_n)$.

The purpose of this exercise is to study sequences of positive terms whose first term $u_0$ is strictly greater than 1 and possessing the following property: for every natural number $n > 0$, the sum of the first $n$ consecutive terms equals the product of the first $n$ consecutive terms. We admit that such a sequence exists and we denote it $(u_n)$. It therefore satisfies three properties:

• $u_0 > 1$,
• for all $n \geqslant 0, u_n \geqslant 0$,
• for all $n > 0, u_0 + u_1 + \cdots + u_{n-1} = u_0 \times u_1 \times \cdots \times u_{n-1}$.

1. We choose $u_0 = 3$. Determine $u_1$ and $u_2$.
2. For every integer $n > 0$, we denote $s_n = u_0 + u_1 + \cdots + u_{n-1} = u_0 \times u_1 \times \cdots \times u_{n-1}$.
   In particular, $s_1 = u_0$. a. Verify that for every integer $n > 0, s_{n+1} = s_n + u_n$ and $s_n > 1$. b. Deduce that for every integer $n > 0$, $$u_n = \frac{s_n}{s_n - 1}$$ c. Show that for all $n \geqslant 0, u_n > 1$.
3. Using the algorithm opposite, we want to calculate the term $u_n$ for a given value of $n$. a. Copy and complete the processing part of the algorithm:
   

   Input:	Enter $n$
   	Enter $u$
   Processing:	$s$ takes the value $u$
   	For $i$ going from 1 to $n$:
   	$u$ takes the value $\ldots$
   	$s$ takes the value $\ldots$
   	End For
   Output:	Display $u$

   
   b. The table below gives values rounded to the nearest thousandth of $u_n$ for different values of the integer $n$:
   

   $n$	0	5	10	20	30	40
   $u_n$	3	1.140	1.079	1.043	1.030	1.023

   
   What conjecture can be made about the convergence of the sequence $(u_n)$?
4. a. Justify that for every integer $n > 0, s_n > n$. b. Deduce the limit of the sequence $(s_n)$ then that of the sequence $(u_n)$.

Exercise 4 -- 5 points -- For candidates who have not followed the specialization course

We study a model of virus propagation in a population, week after week. Each individual in the population can be:

• either susceptible to being affected by the virus (``of type S'');
• either sick (affected by the virus);
• either immunized (cannot be affected by the virus).

For any natural integer $n$, the model of virus propagation is defined by the following rules:

• Among individuals of type S in week $n$, in week $n+1$: $85\%$ remain of type S, $5\%$ become sick and $10\%$ become immunized;
• Among sick individuals in week $n$, in week $n+1$: $65\%$ remain sick, and $35\%$ are cured and become immunized.
• Any individual immunized in week $n$ remains immunized in week $n+1$.

We randomly choose an individual from the population. We consider the following events: $S_{n}$: ``the individual is of type S in week $n$''; $M_{n}$: ``the individual is sick in week $n$''; $I_{n}$: ``the individual is immunized in week $n$''. In week 0, all individuals are considered ``of type S'', so: $$P(S_{0}) = 1 ; \quad P(M_{0}) = 0 \quad \text{and} \quad P(I_{0}) = 0.$$

Part A

We study the evolution of the epidemic during weeks 1 and 2.

1. Reproduce and complete the probability tree.
2. Show that $P(I_{2}) = 0.2025$.
3. Given that an individual is immunized in week 2, what is the probability, rounded to the nearest thousandth, that he was sick in week 1?

Part B

We study the long-term evolution of the disease. For any natural integer $n$, we have: $u_{n} = P(S_{n})$, $v_{n} = P(M_{n})$ and $w_{n} = P(I_{n})$.

Exercise 5 (5 points) — Candidates who have NOT followed the specialization course
A biologist wishes to study the evolution of the population of an animal species in a reserve. This population is estimated at 12000 individuals in 2016. The constraints of the natural environment mean that the population cannot exceed 60000 individuals.

Part A: a first model

In a first approach, the biologist estimates that the population grows by $5\%$ per year. The annual evolution of the population is thus modelled by a sequence $(v_n)$ where $v_n$ represents the number of individuals, expressed in thousands, in $2016 + n$. We thus have $v_0 = 12$.

1. Determine the nature of the sequence $(v_n)$ and give the expression of $v_n$ as a function of $n$.
2. Does this model meet the constraints of the natural environment?

Part B: a second model

The biologist then models the annual evolution of the population by a sequence $(u_n)$ defined by $u_0 = 12$ and, for every natural integer $n$, $$u_{n+1} = -\frac{1.1}{605}u_n^2 + 1.1\, u_n.$$

1. We consider the function $g$ defined on $\mathbb{R}$ by $$g(x) = -\frac{1.1}{605}x^2 + 1.1\, x$$ a. Justify that $g$ is increasing on $[0;60]$. b. Solve in $\mathbb{R}$ the equation $g(x) = x$.
   
2. We note that $u_{n+1} = g(u_n)$. a. Calculate the value rounded to $10^{-3}$ of $u_1$. Interpret. b. Prove by induction that, for every natural integer $n$, $0 \leqslant u_n \leqslant 55$. c. Prove that the sequence $(u_n)$ is increasing. d. Deduce the convergence of the sequence $(u_n)$. e. It is admitted that the limit $\ell$ of the sequence $(u_n)$ satisfies $g(\ell) = \ell$. Deduce its value and interpret it in the context of the exercise.
   
3. The biologist wishes to determine the number of years after which the population will exceed 50000 individuals with this second model. He uses the following algorithm:
   

   \multirow{2}{*}{Variables}	$n$ a natural integer
   \cline{2-2}	$u$ a real number
   Processing	$n$ takes the value 0
   	$u$ takes the value 12
   	While $\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots$
   	$\quad u$ takes the value $\ldots\ldots\ldots\ldots\ldots$
   $n$ takes the value $\ldots\ldots\ldots\ldots\ldots$
   	End While
   Output	Display $\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots$

   
   Copy and complete this algorithm so that it displays as output the smallest integer $r$ such that $u_r \geqslant 50$.

Several transmission lines are assembled end to end, and we assume they introduce errors independently of one another. Each line transmits a bit correctly with probability $p$, and incorrectly with probability $1 - p$. We study the transmission of a single bit, which has value 1 at the beginning of transmission. After passing through $n$ transmission lines, we denote:

• $p _ { n }$ the probability that the received bit has value 1;
• $q _ { n }$ the probability that the received bit has value 0.

We therefore have $p _ { 0 } = 1$ and $q _ { 0 } = 0$. We define the following matrices:
$$A = \left( \begin{array} { c c } p & 1 - p \\ 1 - p & p \end{array} \right) \quad X _ { n } = \binom { p _ { n } } { q _ { n } } \quad P = \left( \begin{array} { c c } 1 & 1 \\ 1 & - 1 \end{array} \right) .$$
We admit that, for every integer $n$, we have: $X _ { n + 1 } = A X _ { n }$ and therefore, $X _ { n } = A ^ { n } X _ { 0 }$.

1. a. Show that $P$ is invertible and determine $P ^ { - 1 }$. b. We set: $D = \left( \begin{array} { c c } 1 & 0 \\ 0 & 2 p - 1 \end{array} \right)$. Verify that: $A = P D P ^ { - 1 }$. c. Show that, for every integer $n \geqslant 1$, $$A ^ { n } = P D ^ { n } P ^ { - 1 } .$$ d. Using the screenshot of a computer algebra system given below, determine the expression of $q _ { n }$ as a function of $n$.
   

   1	$X 0 : = [ [ 1 ] , [ 0 ] ]$
   	$\left[ \begin{array} { l } 1 \\ 0 \end{array} \right]$	M
   2	$\mathrm { P } : = [ [ 1,1 ] , [ 1 , - 1 ] ]$
   	$\left[ \begin{array} { c c } 1 & 1 \\ 1 & - 1 \end{array} \right]$	M
   3	$\mathrm { D } : = [ [ 1,0 ] , [ 0,2 * p - 1 ] ]$
   	$\left[ \begin{array} { c c } 1 & 0 \\ 0 & 2 * p - 1 \end{array} \right]$	M
   4	$P * \left( D ^ { \wedge } n \right) * P ^ { \wedge } ( - 1 ) * X 0$
   	$\left[ \frac { ( 2 * p - 1 ) ^ { n } + 1 } { 2 } \right]$
   	$\frac { - ( 2 * p - 1 ) ^ { n } + 1 } { 2 }$	M

   
   
2. We assume in this question that $p$ equals 0.98. We recall that the bit before transmission has value 1. We wish the probability that the received bit has value 0 be less than or equal to 0.25. What is the maximum number of such transmission lines that can be aligned?

Exercise 5 (5 points) — Candidates who have not followed the speciality course
Let $( u _ { n } )$ be the sequence defined by $u _ { 0 } = 3 , u _ { 1 } = 6$ and, for all natural integer $n$:
$$u _ { n + 2 } = \frac { 5 } { 4 } u _ { n + 1 } - \frac { 1 } { 4 } u _ { n } .$$
The purpose of this exercise is to study the possible limit of the sequence $( u _ { n } )$.

Part A:

We wish to calculate the values of the first terms of the sequence $( u _ { n } )$ using a spreadsheet. We have reproduced below part of a spreadsheet, where the values of $u _ { 0 }$ and $u _ { 1 }$ appear.

	A	B
1	$n$	$u _ { n }$
2	0	3
3	1	6
4	2
5	3
6	4
7	5

1. Give a formula which, entered in cell B4, then copied downwards, allows obtaining values of the sequence $( u _ { n } )$ in column B.
2. Copy and complete the table above. Approximate values to $10 ^ { - 3 }$ of $u _ { n }$ will be given for $n$ ranging from 2 to 5.
3. What can be conjectured about the convergence of the sequence $( u _ { n } )$?

Part B: Study of the sequence

We consider the sequences $( v _ { n } )$ and $( w _ { n } )$ defined for all natural integer $n$ by:
$$v _ { n } = u _ { n + 1 } - \frac { 1 } { 4 } u _ { n } \quad \text { and } \quad w _ { n } = u _ { n } - 7 .$$

1. a. Prove that $( v _ { n } )$ is a constant sequence. b. Deduce that, for all natural integer $n , u _ { n + 1 } = \frac { 1 } { 4 } u _ { n } + \frac { 21 } { 4 }$.
2. a. Using the result of question 1. b., show by induction that, for all natural integer $n , u _ { n } < u _ { n + 1 } < 15$. b. Deduce that the sequence $( u _ { n } )$ is convergent.
3. a. Prove that $( w _ { n } )$ is a geometric sequence and specify its first term and common ratio. b. Deduce that, for all natural integer $n , u _ { n } = 7 - \left( \frac { 1 } { 4 } \right) ^ { n - 1 }$. c. Calculate the limit of the sequence $( u _ { n } )$.

A customer is chosen at random from those who bought a melon during week 1. Among customers who buy a melon in a given week, $90\%$ of them buy a melon the following week; among customers who do not buy a melon in a given week, $60\%$ of them do not buy a melon the following week. For $n \geqslant 1$, we set $p_n = P(A_n)$, where $A_n$ is the event ``the customer buys a melon during week $n$''. Thus $p_1 = 1$. Prove that, for all integer $n \geqslant 1$: $p_{n+1} = 0{,}5\, p_n + 0{,}4$.

For $n \geqslant 1$, we set $p_n = P(A_n)$, where $A_n$ is the event ``the customer buys a melon during week $n$'', with $p_1 = 1$ and $p_{n+1} = 0{,}5\, p_n + 0{,}4$ for all $n \geqslant 1$. a. Prove by induction that, for all integer $n \geqslant 1$: $p_n > 0{,}8$. b. Prove that the sequence $(p_n)$ is decreasing. c. Is the sequence $(p_n)$ convergent?

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 $(p_n)$.

Exercise 4 — Candidates who have followed the specialization course
We call Fibonacci sequence the sequence $( u _ { n } )$ defined by $u _ { 0 } = 0 , u _ { 1 } = 1$ and, for every natural integer $n$,
$$u _ { n + 2 } = u _ { n + 1 } + u _ { n }$$
We admit that, for every natural integer $n$, $u _ { n }$ is a natural integer. Parts A and B can be treated independently.

Part A

1. a. Calculate the terms of the Fibonacci sequence up to $u _ { 10 }$. b. What can be conjectured about the GCD of $u _ { n }$ and $u _ { n + 1 }$ for every natural integer $n$?
2. We define the sequence $\left( v _ { n } \right)$ by $v _ { n } = u _ { n } ^ { 2 } - u _ { n + 1 } \times u _ { n - 1 }$ for every non-zero natural integer $n$. a. Prove that, for every non-zero natural integer $n$, $v _ { n + 1 } = - v _ { n }$. b. Deduce that, for every non-zero natural integer $n$,

$$u _ { n } ^ { 2 } - u _ { n + 1 } \times u _ { n - 1 } = ( - 1 ) ^ { n - 1 }$$
c. Then prove the conjecture made in question 1.b.

Part B

We consider the matrix $F = \left( \begin{array} { l l } 1 & 1 \\ 1 & 0 \end{array} \right)$.

1. Calculate $F ^ { 2 }$ and $F ^ { 3 }$. You may use a calculator.
2. Prove by induction that, for every non-zero natural integer $n$,

$$F ^ { n } = \left( \begin{array} { c c } u_{n+1} & u_n \\ u_n & u_{n-1} \end{array} \right)$$

Exercise 4 (5 points)

Candidates who followed the specialization course

We denote $u_n$ as the number of voles and $v_n$ as the number of foxes on July $1^{\text{st}}$ of the year $2012 + n$.

Part A - A simple model

We model the evolution of populations using the following relations: $$\left\{\begin{array}{l} u_{n+1} = 1{,}1\, u_n - 2000\, v_n \\ v_{n+1} = 2 \times 10^{-5}\, u_n + 0{,}6\, v_n \end{array}\right. \quad \text{for all integers } n \geqslant 0, \text{ with } u_0 = 2000000 \text{ and } v_0 = 120.$$

1. a. We consider the column matrix $U_n = \binom{u_n}{v_n}$ for all integers $n \geqslant 0$.
   Determine the matrix $A$ such that $U_{n+1} = A \times U_n$ for all integers $n$ and give the matrix $U_0$. b. Calculate the number of voles and foxes estimated using this model on July $1^{\text{st}}$ 2018.
2. Let the matrices $P = \left(\begin{array}{cc} 20000 & 5000 \\ 1 & 1 \end{array}\right)$, $D = \left(\begin{array}{cc} 1 & 0 \\ 0 & 0{,}7 \end{array}\right)$ and $P^{-1} = \dfrac{1}{15000}\left(\begin{array}{cc} 1 & -5000 \\ -1 & 20000 \end{array}\right)$.
   We admit that $P^{-1}$ is the inverse matrix of matrix $P$ and that $A = P \times D \times P^{-1}$. a. Show that for all natural integers $n$, $U_n = P \times D^n \times P^{-1} \times U_0$. b. Give without justification the expression of matrix $D^n$ as a function of $n$. c. We admit that, for all natural integers $n$: $$\left\{\begin{array}{lcl} u_n & = & \dfrac{2{,}8 \times 10^7 + 2 \times 10^6 \times 0{,}7^n}{15} \\[6pt] v_n & = & \dfrac{1400 + 400 \times 0{,}7^n}{15} \end{array}\right.$$ Describe the evolution of the two populations.

Part B - A model more in line with reality

We construct another model using the following relations: $$\left\{\begin{array}{l} u_{n+1} = 1{,}1\, u_n - 0{,}001\, u_n \times v_n \\ v_{n+1} = 2 \times 10^{-7}\, u_n \times v_n + 0{,}6\, v_n \end{array}\right. \quad \text{for all integers } n \geqslant 0, \text{ with } u_0 = 2000000 \text{ and } v_0 = 120.$$

1. What formulas must be written in cells B4 and C4 and copied downwards to fill columns B and C?
2. With the second model, from what year do we observe the phenomenon described (decrease in foxes and increase in voles)?

Part C

In this part we use the model from Part B. Is it possible to give $u_0$ and $v_0$ values such that the two populations remain stable from one year to the next, that is, such that for all natural integers $n$ we have $u_{n+1} = u_n$ and $v_{n+1} = v_n$? (We then speak of a stable state.)

Exercise 5

5 points

Candidates who have not followed the specialized course

Let $k$ be a strictly positive real number. We consider the sequence ( $u _ { n }$ ) defined by $u _ { 0 } = 1 , u _ { 1 } = k$ and, for all natural integer $n$ by:
$$u _ { n + 2 } = \frac { u _ { n + 1 } ^ { 2 } } { k u _ { n } }$$
It is admitted that all terms of the sequence ( $u _ { n }$ ) exist and are strictly positive.

1. Express $u _ { 2 } , u _ { 3 }$ and $u _ { 4 }$ as functions of $k$.
2. Using a spreadsheet, the first terms of the sequence ( $u _ { n }$ ) were calculated for two values of $k$. The value of the real number $k$ is entered in cell E 2 .

	A	B	C	D	E		A	B	C	D	E
1	$n$	$u ( n )$				1	$n$	$u ( n )$
2	0	1		$k =$	2.7182818	2	0	1		$k =$	0.9
3	1	2.7182818				3	1	0.9
4	2	2.7182818				4	2	0.9
5	3	1				5	3	1
6	4	0.1353353				6	4	1.2345679
7	5	0.0067319				7	5	1.6935088
8	6	0.000 1234				8	6	2.581 1748
9	7	$8.315 \mathrm { E } - 07$				9	7	4.3712422
10	8	$2.061 \mathrm { E } - 09$				10	8	8.2252633

The purpose of this exercise is to study the sequence $(u_n)$ defined by the value of its first term $u_1$ and, for every natural number $n$ greater than or equal to 1, by the relation: $$u_{n+1} = (n+1)u_n - 1$$
Part A

1. Verify, by detailing the calculation, that if $u_1 = 0$ then $u_4 = -17$.
2. Copy and complete the algorithm below so that by first entering in $U$ a value of $u_1$ it calculates the terms of the sequence $(u_n)$ from $u_2$ to $u_{13}$.
   For $N$ going from 1 to 12 $$U \leftarrow$$ End For
   
3. This algorithm was executed for $u_1 = 0.7$ then for $u_1 = 0.8$.
   Here are the values obtained.
   

   For $u_1 = 0.7$	For $u_1 = 0.8$
   0.4	0.6
   0.2	0.8
   -0.2	2.2
   -2	10
   -13	59
   -92	412
   -737	3295
   -6634	29654
   -66341	296539
   -729752	3261928
   -8757025	39143135
   -113841326	508860754

   
   What appears to be the limit of this sequence if $u_1 = 0.7$? And if $u_1 = 0.8$?

Part B
We consider the sequence $(I_n)$ defined for every natural number $n$, greater than or equal to 1, by: $$I_n = \int_0^1 x^n \mathrm{e}^{1-x} \mathrm{~d}x$$ We recall that the number e is the value of the exponential function at 1, that is to say that $\mathrm{e} = \mathrm{e}^1$.

1. Prove that the function $F$ defined on the interval $[0;1]$ by $F(x) = (-1-x)\mathrm{e}^{1-x}$ is an antiderivative on the interval $[0;1]$ of the function $f$ defined on the interval $[0;1]$ by $f(x) = x\mathrm{e}^{1-x}$.
2. Deduce that $I_1 = \mathrm{e} - 2$.
3. It is admitted that, for every natural number $n$ greater than or equal to 1, we have: $$I_{n+1} = (n+1)I_n - 1.$$ Use this formula to calculate $I_2$.
4. a. Justify that, for every real number $x$ in the interval $[0;1]$ and for every natural number $n$ greater than or equal to 1, we have: $0 \leqslant x^n \mathrm{e}^{1-x} \leqslant x^n \mathrm{e}$. b. Justify that: $\int_0^1 x^n \mathrm{e} \, \mathrm{d}x = \dfrac{\mathrm{e}}{n+1}$. c. Deduce that, for every natural number $n$ greater than or equal to 1, we have: $0 \leqslant I_n \leqslant \dfrac{\mathrm{e}}{n+1}$. d. Determine $\lim_{n \rightarrow +\infty} I_n$.

Part C
In this part, we denote by $n!$ the number defined, for every natural number $n$ greater than or equal to 1, by: $1! = 1$, $2! = 2 \times 1$, and if $n \geqslant 3$: $n! = n \times (n-1) \times \ldots \times 1$. And, more generally: $(n+1)! = (n+1) \times n!$

1. Prove by induction that, for every natural number $n$ greater than or equal to 1, we have: $$u_n = n! \left(u_1 - \mathrm{e} + 2\right) + I_n$$ We recall that, for every natural number $n$ greater than or equal to 1, we have: $$u_{n+1} = (n+1)u_n - 1 \quad \text{and} \quad I_{n+1} = (n+1)I_n - 1.$$
2. It is admitted that: $\lim_{n \rightarrow +\infty} n! = +\infty$. a. Determine the limit of the sequence $(u_n)$ when $u_1 = 0.7$. b. Determine the limit of the sequence $(u_n)$ when $u_1 = 0.8$.

Part A: establishing an inequality
On the interval $[ 0 ; + \infty [$, we define the function $f$ by $f ( x ) = x - \ln ( x + 1 )$.

1. Study the monotonicity of the function $f$ on the interval $[ 0 ; + \infty [$.
2. Deduce that for all $x \in [ 0 ; + \infty [ , \ln ( x + 1 ) \leqslant x$.

Part B: application to the study of a sequence
We set $u _ { 0 } = 1$ and for every natural number $n , u _ { n + 1 } = u _ { n } - \ln \left( 1 + u _ { n } \right)$. We admit that the sequence with general term $u _ { n }$ is well defined.

1. Calculate an approximate value to $10 ^ { - 3 }$ of $u _ { 2 }$.
2. a. Prove by induction that for every natural number $n , \quad u _ { n } \geqslant 0$. b. Prove that the sequence $( u _ { n } )$ is decreasing, and deduce that for every natural number $n , \quad u _ { n } \leqslant 1$. c. Show that the sequence $( u _ { n } )$ is convergent.
3. Let $\ell$ denote the limit of the sequence $( u _ { n } )$ and we admit that $\ell = f ( \ell )$, where $f$ is the function defined in Part A. Deduce the value of $\ell$.
4. a. Write an algorithm which, for a given natural number $p$, allows us to determine the smallest rank $N$ from which all terms of the sequence $( u _ { n } )$ are less than $10 ^ { - p }$. b. Determine the smallest natural number $n$ from which all terms of the sequence $( u _ { n } )$ are less than $10 ^ { - 15 }$.

We consider the sequence $\left(u_n\right)$ defined for every integer $n \geqslant 0$ by: $\left\{\begin{array}{l}u_{n+1} = 3 - \dfrac{10}{u_n + 4}\\u_0 = 5\end{array}\right.$

Part A:

1. Determine the exact value of $u_1$ and $u_2$.
2. Prove by induction that for every natural integer $n$, $u_n \geqslant 1$.
3. Prove that, for every natural integer $n$, $u_{n+1} - u_n = \dfrac{\left(1 - u_n\right)\left(u_n + 2\right)}{u_n + 4}$.
4. Deduce the direction of variation of the sequence $(u_n)$.
5. Justify that the sequence $\left(u_n\right)$ converges.

Part B:

We consider the sequence $(v_n)$ defined for every natural integer $n$ by $v_n = \dfrac{u_n - 1}{u_n + 2}$.

1. a. Prove that $\left(v_n\right)$ is a geometric sequence whose common ratio and first term $v_0$ we will determine. b. Express $v_n$ as a function of $n$.
   Deduce that for every natural integer $n$, $v_n \neq 1$.
2. Prove that for every natural integer $n$, $u_n = \dfrac{2v_n + 1}{1 - v_n}$.
3. Deduce the limit of the sequence $(u_n)$.

Part C:

We consider the algorithm below.

$u \leftarrow 5$

$n \leftarrow 0$

While $u \geqslant 1.01$

$n \leftarrow n + 1$

$u \leftarrow 3 - \dfrac{10}{u + 4}$

End While

1. After execution of the algorithm, what value is contained in the variable $n$?
2. Using parts A and B, interpret this value.

For candidates who have not followed the specialization course
Let $f$ be the function defined on the interval $[ 0 ; 4]$ by $$f ( x ) = \frac { 2 + 3 x } { 4 + x }$$

Part A

We consider the sequence ( $u _ { n }$ ) defined by: $$u _ { 0 } = 3 \text { and for all natural integer } n , u _ { n + 1 } = f \left( u _ { n } \right) .$$ It is admitted that this sequence is well defined.

1. Calculate $u _ { 1 }$.
2. Show that the function $f$ is increasing on the interval $[ 0 ; 4 ]$.
3. Show that for all natural integer $n$, $$1 \leqslant u _ { n + 1 } \leqslant u _ { n } \leqslant 3$$
4. a. Show that the sequence ( $u _ { n }$ ) is convergent. b. We call $\ell$ the limit of the sequence ( $u _ { n }$ ); show the equality: $$\ell = \frac { 2 + 3 \ell } { 4 + \ell }$$ c. Determine the value of the limit $\ell$.

Part B

We consider the sequence $\left( v _ { n } \right)$ defined by: $$v _ { 0 } = 0.1 \text { and for all natural integer } n , v _ { n + 1 } = f \left( v _ { n } \right) .$$

1. We give in the Annex the representative curve $\mathscr { C } _ { f }$ of the function $f$ and the line $D$ with equation $y = x$. Place on the $x$-axis by geometric construction the terms $v _ { 1 } , v _ { 2 }$ and $v _ { 3 }$ on the annex, to be returned with the copy. What conjecture can be formulated about the direction of variation and the behavior of the sequence ( $v _ { n }$ ) as $n$ tends to infinity?
2. a. Show that for all natural integer $n$, $$1 - v _ { n + 1 } = \left( \frac { 2 } { 4 + v _ { n } } \right) \left( 1 - v _ { n } \right)$$ b. Show by induction that for all natural integer $n$, $$0 \leqslant 1 - v _ { n } \leqslant \left( \frac { 1 } { 2 } \right) ^ { n }$$
3. Does the sequence $\left( v _ { n } \right)$ converge? If so, specify its limit.

For candidates who have followed the specialization course
The two parts are independent.

Part A

A laboratory studies the evolution of a population of parasitic insects on plants. This evolution has two stages: a larval stage and an adult stage which is the only one during which insects can reproduce. Observation of the evolution of this population leads to proposing the following model. Each week:

• Each adult gives birth to 2 larvae then $75\%$ of adults die.
• $25\%$ of larvae die and $50\%$ of larvae become adults.

For all natural integer $n$, we denote $\ell _ { n }$ the number of larvae and $a _ { n }$ the number of adults after $n$ weeks. For all natural integer $n$, we denote $X _ { n }$ the column matrix defined by: $X _ { n } = \binom { \ell _ { n } } { a _ { n } }$

1. Show that, for all natural integer $n$, $X _ { n + 1 } = A X _ { n }$ where $A$ is the matrix: $$A = \left( \begin{array} { c c } 0.25 & 2 \\ 0.5 & 0.25 \end{array} \right)$$
2. We denote $U$ and $V$ the column matrices: $U = \binom { 2 } { 1 }$ and $V = \binom { a } { 1 }$, where $a$ is a real number. a. Show that $A U = 1.25 U$. b. Determine the real number $a$ such that $A V = - 0.75 V$.

In questions 3 and 4, the real number $a$ is fixed so that it is the solution of $A V = - 0.75 V$.

1. It is admitted that there exist two real numbers $\alpha$ and $\beta$ such that: $X _ { 0 } = \alpha U + \beta V$ and $\alpha > 0$. a. Show that, for all natural integer $n$, $X _ { n } = \alpha ( 1.25 ) ^ { n } U + \beta ( - 0.75 ) ^ { n } V$. b. Deduce that for all natural integer $n$: $$\left\{ \begin{array} { l } \ell _ { n } = 2 ( 1.25 ) ^ { n } \left( \alpha - \beta ( - 0.6 ) ^ { n } \right) \\ a _ { n } = ( 1.25 ) ^ { n } \left( \alpha + \beta ( - 0.6 ) ^ { n } \right) . \end{array} \right.$$
2. Show that $\lim _ { n \rightarrow + \infty } \frac { \ell _ { n } } { a _ { n } } = 2$. Interpret this result in the context of the exercise.

Part B

1. We consider the equation $( E ) : 19 x - 6 y = 1$. Determine the number of couples of integers ( $x ; y$ ) solutions of the equation $( E )$ and satisfying $2000 \leqslant x \leqslant 2100$.
2. Let $n$ be a natural integer. Show that the integers ( $2 n + 3$ ) and ( $n + 3$ ) are coprime if and only if $n$ is not a multiple of 3.

The complex plane is equipped with a direct orthonormal coordinate system ( $\mathrm { O } ; \vec { u } , \vec { v }$ ). Consider the sequence of complex numbers ( $z _ { n }$ ) defined by: $$z _ { 0 } = 0 \text { and for every natural number } n , z _ { n + 1 } = ( 1 + \mathrm { i } ) z _ { n } - \mathrm { i }$$ For every natural number $n$, let $A _ { n }$ denote the point with affix $z _ { n }$. Let B denote the point with affix 1.

1. a. Show that $z _ { 1 } = - \mathrm { i }$ and that $z _ { 2 } = 1 - 2 \mathrm { i }$. b. Calculate $z _ { 3 }$. c. On your answer sheet, plot the points $\mathrm { B } , A _ { 1 } , A _ { 2 }$ and $A _ { 3 }$ in the direct orthonormal coordinate system $( \mathrm { O } ; \vec { u } , \vec { v } )$. d. Prove that the triangle $\mathrm { B } A _ { 1 } A _ { 2 }$ is isosceles right-angled.
2. For every natural number $n$, set $u _ { n } = \left| z _ { n } - 1 \right|$. a. Prove that for every natural number $n$, we have $u _ { n + 1 } = \sqrt { 2 } u _ { n }$. b. Determine from which natural number $n$ the distance $\mathrm { B } A _ { n }$ is strictly greater than 1000. Detail the approach chosen.
3. a. Determine the exponential form of the complex number $1 + \mathrm { i }$. b. Prove by induction that for every natural number $n$, $z _ { n } = 1 - ( \sqrt { 2 } ) ^ { n } \mathrm { e } ^ { \mathrm { i } \frac { n \pi } { 4 } }$. c. Does the point $A _ { 2020 }$ belong to the x-axis? Justify.