Which of the following series is conditionally convergent?
(A) $\sum _ { k = 1 } ^ { \infty } ( - 1 ) ^ { k } \frac { 5 } { k ^ { 3 } + 1 }$
(B) $\sum _ { k = 1 } ^ { \infty } ( - 1 ) ^ { k } \frac { 5 } { k + 1 }$
(C) $\sum _ { k = 1 } ^ { \infty } ( - 1 ) ^ { k } \frac { 5 k } { k + 1 }$
(D) $\sum _ { k = 1 } ^ { \infty } ( - 1 ) ^ { k } \frac { 5 k ^ { 2 } } { k + 1 }$

Which of the following series converge?
I. $\sum _ { n = 1 } ^ { \infty } \frac { 8 ^ { n } } { n ! }$
II. $\sum _ { n = 1 } ^ { \infty } \frac { n ! } { n ^ { 100 } }$
III. $\sum _ { n = 1 } ^ { \infty } \frac { n + 1 } { ( n ) ( n + 2 ) ( n + 3 ) }$
(A) I only
(B) II only
(C) III only
(D) I and III only
(E) I, II, and III

What is the radius of convergence of the series $\sum _ { n = 0 } ^ { \infty } \frac { ( x - 4 ) ^ { 2 n } } { 3 ^ { n } }$ ?
(A) $2 \sqrt { 3 }$
(B) 3
(C) $\sqrt { 3 }$
(D) $\frac { \sqrt { 3 } } { 2 }$
(E) 0

The power series $\sum _ { n = 0 } ^ { \infty } a _ { n } ( x - 3 ) ^ { n }$ converges at $x = 5$. Which of the following must be true?
(A) The series diverges at $x = 0$.
(B) The series diverges at $x = 1$.
(C) The series converges at $x = 1$.
(D) The series converges at $x = 2$.
(E) The series converges at $x = 6$.

For what values of $p$ will both series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 2 p } }$ and $\sum _ { n = 1 } ^ { \infty } \left( \frac { p } { 2 } \right) ^ { n }$ converge?
(A) $- 2 < p < 2$ only
(B) $- \frac { 1 } { 2 } < p < \frac { 1 } { 2 }$ only
(C) $\frac { 1 } { 2 } < p < 2$ only
(D) $p < \frac { 1 } { 2 }$ and $p > 2$
(E) There are no such values of $p$.

If the series $\sum _ { n = 1 } ^ { \infty } a _ { n }$ converges and $a _ { n } > 0$ for all $n$, which of the following must be true?
(A) $\lim _ { n \rightarrow \infty } \left| \frac { a _ { n + 1 } } { a _ { n } } \right| = 0$
(B) $\left| a _ { n } \right| < 1$ for all $n$
(C) $\sum _ { n = 1 } ^ { \infty } a _ { n } = 0$
(D) $\sum _ { n = 1 } ^ { \infty } n a _ { n }$ diverges.
(E) $\sum _ { n = 1 } ^ { \infty } \frac { a _ { n } } { n }$ converges.

We define the sequences $(u_n)$ and $(v_n)$ on the set $\mathbb{N}$ of natural numbers by: $$u_0 = 0 ; v_0 = 1, \text{ and } \left\{\begin{array}{l} u_{n+1} = \dfrac{u_n + v_n}{2} \\ v_{n+1} = \dfrac{u_n + 2v_n}{3} \end{array}\right.$$
The purpose of this exercise is to study the convergence of sequences $(u_n)$ and $(v_n)$.

1. Calculate $u_1$ and $v_1$.
2. We consider the following algorithm:
   Variables: $u$, $v$ and $w$ real numbers; $N$ and $k$ integers Initialization: $u$ takes the value 0; $v$ takes the value 1 Start of algorithm Enter the value of $N$ For $k$ varying from 1 to $N$ $w$ takes the value $u$ $u$ takes the value $\dfrac{w + v}{2}$ $v$ takes the value $\dfrac{w + 2v}{3}$ End of For Display $u$ Display $v$ End of algorithm
   a. We execute this algorithm by entering $N = 2$. Copy and complete the table given below containing the state of variables during the execution of the algorithm.
   

   $k$	$w$	$u$	$v$
   1
   2

   
   b. For a given number $N$, what do the values displayed by the algorithm correspond to with respect to the situation studied in this exercise?
   
3. For all natural numbers $n$ we define the column vector $X_n$ by $X_n = \binom{u_n}{v_n}$ and the matrix $A$ by $$A = \left(\begin{array}{ll} \dfrac{1}{2} & \dfrac{1}{2} \\ \dfrac{1}{3} & \dfrac{2}{3} \end{array}\right).$$ a. Verify that, for all natural numbers $n$, $X_{n+1} = A X_n$. b. Prove by induction that $X_n = A^n X_0$ for all natural numbers $n$.
   
4. We define matrices $P$, $P'$ and $B$ by $$P = \left(\begin{array}{cc} \dfrac{4}{5} & \dfrac{6}{5} \\ -\dfrac{6}{5} & \dfrac{6}{5} \end{array}\right), \quad P' = \left(\begin{array}{cc} \dfrac{1}{2} & -\dfrac{1}{2} \\ \dfrac{1}{2} & \dfrac{1}{3} \end{array}\right)$$

Exercise 2 — Candidates WHO HAVE NOT FOLLOWED the mathematics specialization course
We consider the sequence $( u _ { n } )$ defined by $u _ { 0 } = 1$ and, for every natural integer $n$, $$u _ { n + 1 } = \sqrt { 2 u _ { n } } .$$

1. We consider the following algorithm:

Variables:	$n$ is a natural integer
	$u$ is a positive real number
Initialization:	Request the value of $n$
	Assign to $u$ the value 1
Processing:	For $i$ varying from 1 to $n :$
	$\mid$ Assign to $u$ the value $\sqrt { 2 u }$
	End of For
Output :	Display $u$

a. Give an approximate value to $10 ^ { - 4 }$ of the result displayed by this algorithm when $n = 3$ is chosen. b. What does this algorithm allow us to calculate? c. The table below gives approximate values obtained using this algorithm for certain values of $n$.

$n$	1	5	10	15	20
Displayed value	1,4142	1,9571	1,9986	1,9999	1,9999

What conjectures can be made concerning the sequence $\left( u _ { n } \right)$ ?
2. a. Prove that, for every natural integer $n , 0 < u _ { n } \leqslant 2$. b. Determine the direction of variation of the sequence $( u _ { n } )$. c. Prove that the sequence $( u _ { n } )$ is convergent. We do not ask for the value of its limit.
3. We consider the sequence $( v _ { n } )$ defined, for every natural integer $n$, by $v _ { n } = \ln u _ { n } - \ln 2$. a. Prove that the sequence $\left( v _ { n } \right)$ is a geometric sequence with ratio $\frac { 1 } { 2 }$ and first term $v _ { 0 } = - \ln 2$. b. Determine, for every natural integer $n$, the expression of $v _ { n }$ as a function of $n$, then of $u _ { n }$ as a function of $n$. c. Determine the limit of the sequence $\left( u _ { n } \right)$. d. Copy the algorithm below and complete it with the instructions for processing and output, so as to display as output the smallest value of $n$ such that $u _ { n } > 1,999$.

Variables:	$n$ is a natural integer
	$u$ is a real number
Initialization :	Assign to $n$ the value 0
	Assign to $u$ the value 1
Processing:
Output :

We study the evolution over time of the number of young and adult animals in a population. For any natural integer $n$, we denote by $j _ { n }$ the number of young animals after $n$ years of observation and $a _ { n }$ the number of adult animals after $n$ years of observation. At the beginning of the first year of the study, there are 200 young animals and 500 adult animals. Thus $j _ { 0 } = 200$ and $a _ { 0 } = 500$. We admit that for any natural integer $n$ we have: $$\left\{ \begin{array} { l } j _ { n + 1 } = 0,125 j _ { n } + 0,525 a _ { n } \\ a _ { n + 1 } = 0,625 j _ { n } + 0,625 a _ { n } \end{array} \right.$$ We introduce the following matrices: $$A = \left( \begin{array} { l l } 0,125 & 0,525 \\ 0,625 & 0,625 \end{array} \right) \text { and, for any natural integer } n , U _ { n } = \binom { j _ { n } } { a _ { n } } .$$

1. a. Show that for any natural integer $n , U _ { n + 1 } = A \times U _ { n }$. b. Calculate the number of young animals and adult animals after one year of observation and then after two years of observation (results rounded down to the nearest unit). c. For any non-zero natural integer $n$, express $U _ { n }$ as a function of $A ^ { n }$ and $U _ { 0 }$.
2. We introduce the following matrices $Q = \left( \begin{array} { c c } 7 & 3 \\ - 5 & 5 \end{array} \right)$ and $D = \left( \begin{array} { c c } - 0,25 & 0 \\ 0 & 1 \end{array} \right)$. a. We admit that the matrix $Q$ is invertible and that $Q ^ { - 1 } = \left( \begin{array} { c c } 0,1 & - 0,06 \\ 0,1 & 0,14 \end{array} \right)$. Show that $Q \times D \times Q ^ { - 1 } = A$. b. Show by induction on $n$ that for any non-zero natural integer $n$: $A ^ { n } = Q \times D ^ { n } \times Q ^ { - 1 }$. c. For any non-zero natural integer $n$, determine $D ^ { n }$ as a function of $n$.
3. We admit that for any non-zero natural integer $n$, $$A ^ { n } = \left( \begin{array} { l l } 0,3 + 0,7 \times ( - 0,25 ) ^ { n } & 0,42 - 0,42 \times ( - 0,25 ) ^ { n } \\ 0,5 - 0,5 \times ( - 0,25 ) ^ { n } & 0,7 + 0,3 \times ( - 0,25 ) ^ { n } \end{array} \right)$$ a. Deduce the expressions of $j _ { n }$ and $a _ { n }$ as functions of $n$ and determine the limits of these two sequences. b. What can we conclude about the population of animals studied?

Exercise 4 (5 points) -- Candidates who have NOT chosen the specialization option
Part A
Consider the sequence $(u_{n})$ defined by: $u_{0} = 2$ and, for all natural integers $n$: $$u_{n+1} = \frac{1 + 3u_{n}}{3 + u_{n}}$$ It is admitted that all terms of this sequence are defined and strictly positive.

1. Prove by induction that, for all natural integers $n$, we have: $u_{n} > 1$.
2. a. Establish that, for all natural integers $n$, we have: $u_{n+1} - u_{n} = \frac{(1 - u_{n})(1 + u_{n})}{3 + u_{n}}$. b. Determine the direction of variation of the sequence $(u_{n})$. Deduce that the sequence $(u_{n})$ converges.

Part B
Consider the sequence $(u_{n})$ defined by: $u_{0} = 2$ and, for all natural integers $n$: $$u_{n+1} = \frac{1 + 0.5u_{n}}{0.5 + u_{n}}$$ It is admitted that all terms of this sequence are defined and strictly positive.

1. Consider the following algorithm:
   

   Input	Let $n$ be a non-zero natural integer
   Initialization	Assign to $u$ the value 2
   Processing	FOR $i$ going from 1 to $n$
   and	Assign to $u$ the value $\frac{1 + 0.5u}{0.5 + u}$
   output	Display $u$
   	END FOR

   
   Reproduce and complete the following table by running this algorithm for $n = 3$. The values of $u$ should be rounded to the nearest thousandth.
   

   $i$	1	2	3
   $u$

   
   
2. For $n = 12$, the previous table was extended and we obtained:
   

   $i$	4	5	6	7	8	9	10	11	12
   $u$	1.0083	0.9973	1.0009	0.9997	1.0001	0.99997	1.00001	0.999996	1.000001

   
   Conjecture the behavior of the sequence $(u_{n})$ at infinity.
3. Consider the sequence $(v_{n})$ defined, for all natural integers $n$, by: $v_{n} = \frac{u_{n} - 1}{u_{n} + 1}$. a. Prove that the sequence $(v_{n})$ is geometric with common ratio $-\frac{1}{3}$. b. Calculate $v_{0}$ then write $v_{n}$ as a function of $n$.
4. a. Show that, for all natural integers $n$, we have: $v_{n} \neq 1$. b. Show that, for all natural integers $n$, we have: $u_{n} = \frac{1 + v_{n}}{1 - v_{n}}$. c. Determine the limit of the sequence $(u_{n})$.

The purpose of this exercise is the study of the sequence $(u_n)$ defined by its first term $u_1 = \frac{3}{2}$ and the recurrence relation: $u_{n+1} = \frac{n u_n + 1}{2(n+1)}$.

Part A - Algorithms and conjectures

To calculate and display the term $u_9$ of the sequence, a student proposes the algorithm below. He forgot to complete two lines.

Variables	$n$ is a natural integer, $u$ is a real number
Initialization	Assign to $n$ the value 1, Assign to $u$ the value 1.5
Treatment	While $n < 9$, Assign to $u$ the value $\cdots$, Assign to $n$ the value $\cdots$, End While
Output	Display the variable $u$

1. Copy and complete the two lines of the algorithm where there are ellipses.
2. How would this algorithm need to be modified so that it calculates and displays all terms of the sequence from $u_2$ to $u_9$?
3. With this modified algorithm, the following results were obtained, rounded to the nearest ten-thousandth:
   

   n	1	2	3	4	5	6	$\ldots$	99	100
   $u_n$	1.5	0.625	0.375	0.2656	0.2063	0.1693	$\ldots$	0.0102	0.0101

   
   In light of these results, conjecture the direction of variation and convergence of the sequence $(u_n)$.

Part B - Mathematical study

We define an auxiliary sequence $(v_n)$ by: for all integer $n \geqslant 1, v_n = n u_n - 1$.

1. Show that the sequence $(v_n)$ is geometric; specify its common ratio and its first term.
2. Deduce that, for all natural integer $n \geqslant 1$, we have: $u_n = \frac{1 + (0.5)^n}{n}$.
3. Determine the limit of the sequence $(u_n)$.
4. Justify that, for all integer $n \geqslant 1$, we have: $u_{n+1} - u_n = -\frac{1 + (1 + 0.5n)(0.5)^n}{n(n+1)}$. Deduce the direction of variation of the sequence $(u_n)$.

Part C - Return to algorithms

Inspired by part A, write an algorithm to determine and display the smallest integer $n$ such that $u_n < 0.001$.

Exercise 4 — Candidates who have NOT followed the specialization course
We consider the sequence $(u_n)$ defined on $\mathbb{N}$ by: $$u_0 = 2 \text{ and for all natural integer } n,\quad u_{n+1} = \frac{u_n + 2}{2u_n + 1}.$$ We admit that for all natural integer $n$, $u_n > 0$.

1. a. Calculate $u_1, u_2, u_3, u_4$. An approximate value to $10^{-2}$ may be given. b. Verify that if $n$ is one of the integers $0,1,2,3,4$ then $u_n - 1$ has the same sign as $(-1)^n$. c. Establish that for all natural integer $n$, $u_{n+1} - 1 = \dfrac{-u_n + 1}{2u_n + 1}$. d. Prove by induction that for all natural integer $n$, $u_n - 1$ has the same sign as $(-1)^n$.
2. For all natural integer $n$, we set $v_n = \dfrac{u_n - 1}{u_n + 1}$. a. Establish that for all natural integer $n$, $v_{n+1} = \dfrac{-u_n + 1}{3u_n + 3}$. b. Prove that the sequence $(v_n)$ is a geometric sequence with ratio $-\dfrac{1}{3}$. Deduce the expression of $v_n$ as a function of $n$. c. We admit that for all natural integer $n$, $u_n = \dfrac{1 + v_n}{1 - v_n}$. Express $u_n$ as a function of $n$ and determine the limit of the sequence $(u_n)$.

(For candidates who have not followed the specialization course) Let the numerical sequence $(u _ { n })$ defined on $\mathbf{N}$ by: $$u _ { 0 } = 2 \quad \text { and for every natural number } n , u _ { n + 1 } = \frac { 2 } { 3 } u _ { n } + \frac { 1 } { 3 } n + 1$$

1. 1. [a.] Calculate $u _ { 1 } , u _ { 2 } , u _ { 3 }$ and $u _ { 4 }$. Approximate values to $10 ^ { - 2 }$ may be given.
   2. [b.] Form a conjecture about the monotonicity of this sequence.
2. 1. [a.] Prove that for every natural number $n$, $$u _ { n } \leqslant n + 3$$
   2. [b.] Prove that for every natural number $n$, $$u _ { n + 1 } - u _ { n } = \frac { 1 } { 3 } \left( n + 3 - u _ { n } \right)$$
   3. [c.] Deduce a validation of the previous conjecture.
3. We denote by $\left( v _ { n } \right)$ the sequence defined on $\mathbf { N }$ by $v _ { n } = u _ { n } - n$.
   1. [a.] Prove that the sequence $\left( v _ { n } \right)$ is a geometric sequence with common ratio $\frac { 2 } { 3 }$.
   2. [b.] Deduce that for every natural number $n$, $$u _ { n } = 2 \left( \frac { 2 } { 3 } \right) ^ { n } + n$$
   3. [c.] Determine the limit of the sequence $(u _ { n })$.
4. For every non-zero natural number $n$, we set: $$S _ { n } = \sum _ { k = 0 } ^ { n } u _ { k } = u _ { 0 } + u _ { 1 } + \ldots + u _ { n } \quad \text { and } \quad T _ { n } = \frac { S _ { n } } { n ^ { 2 } } .$$
   1. [a.] Express $S _ { n }$ as a function of $n$.
   2. [b.] Determine the limit of the sequence $(T _ { n })$.

(For candidates who have followed the specialization course) We study the population of an imaginary region. On January 1, 2013, this region had 250,000 inhabitants, of which $70 \%$ lived in the countryside and $30 \%$ in the city. Examination of statistical data collected over several years leads to the choice of modelling the population evolution for the coming years as follows:

• the total population is globally constant,
• each year, $5 \%$ of those living in the city decide to move to the countryside and $1 \%$ of those living in the countryside choose to move to the city.

For every natural number $n$, we denote $v _ { n }$ the number of inhabitants of this region living in the city on January 1 of the year $(2013 + n)$ and $c _ { n }$ the number of those living in the countryside on the same date.

1. For every natural number $n$, express $v _ { n + 1 }$ and $c _ { n + 1 }$ as functions of $v _ { n }$ and $c _ { n }$.
2. Let the matrix $A = \left( \begin{array} { l l } 0{,}95 & 0{,}01 \\ 0{,}05 & 0{,}99 \end{array} \right)$.

Exercise 4 (Candidates who have not followed the mathematics specialization course)
Consider the sequence $( u _ { n } )$ defined by $u _ { 0 } = \frac { 1 } { 2 }$ and such that for every natural integer $n$,
$$u _ { n + 1 } = \frac { 3 u _ { n } } { 1 + 2 u _ { n } }$$

1. a. Calculate $u _ { 1 }$ and $u _ { 2 }$. b. Prove, by induction, that for every natural integer $n , 0 < u _ { n }$.
2. We admit that for every natural integer $n , u _ { n } < 1$. a. Prove that the sequence $\left( u _ { n } \right)$ is increasing. b. Prove that the sequence $( u _ { n } )$ converges.
3. Let $\left( v _ { n } \right)$ be the sequence defined, for every natural integer $n$, by $v _ { n } = \frac { u _ { n } } { 1 - u _ { n } }$. a. Show that the sequence $\left( v _ { n } \right)$ is a geometric sequence with ratio 3. b. Express for every natural integer $n , v _ { n }$ as a function of $n$. c. Deduce that, for every natural integer $n , u _ { n } = \frac { 3 ^ { n } } { 3 ^ { n } + 1 }$. d. Determine the limit of the sequence $\left( u _ { n } \right)$.

5. Using the previous questions, the following result can be established, which is admitted.
For every non-zero natural integer $n$,
$$A ^ { n } = \left( \begin{array} { c c } - 2 ^ { n + 1 } + 3 ^ { n + 1 } & 3 \times 2 ^ { n + 1 } - 2 \times 3 ^ { n + 1 } \\ - 2 ^ { n } + 3 ^ { n } & 3 \times 2 ^ { n } - 2 \times 3 ^ { n } \end{array} \right)$$
Deduce an expression for $u _ { n }$ as a function of $n$. Does the sequence ( $u _ { n }$ ) have a limit?

APPENDIX for EXERCISE 3, to be returned with the answer sheet

Graphical representation $\mathscr { C } _ { 1 }$ of the function $f _ { 1 }$ [Figure]

For each of the following propositions, indicate whether it is true or false and justify your chosen answer. One point is awarded for each correct answer that is properly justified. An answer without justification is not taken into account. An absence of an answer is not penalized.
Proposition 1 Every positive increasing sequence tends to $+ \infty$.
Proposition 2 $g$ is the function defined on $] - \frac { 1 } { 2 } ; + \infty [$ by $$g ( x ) = 2 x \ln ( 2 x + 1 ) .$$ On $] - \frac { 1 } { 2 } ; + \infty$ [, the equation $g ( x ) = 2 x$ has a unique solution: $\frac { \mathrm { e } - 1 } { 2 }$.
Proposition 3 The slope of the tangent line to the curve representing the function $g$ at the point with abscissa $\frac { 1 } { 2 }$ is: $1 + \ln 4$.
Proposition 4 Space is equipped with an orthonormal coordinate system ( $\mathrm { O } , \vec { \imath } , \vec { \jmath } , \vec { k }$ ). $\mathscr { P }$ and $\mathscr { R }$ are the planes with equations respectively: $2 x + 3 y - z - 11 = 0$ and $x + y + 5 z - 11 = 0$. The planes $\mathscr { P }$ and $\mathscr { R }$ intersect perpendicularly.

Consider the numerical sequence $(u_n)$ defined on $\mathbb{N}$ by:
$$u _ { 0 } = 2 \quad \text { and for every natural number } n , \quad u _ { n + 1 } = - \frac { 1 } { 2 } u _ { n } ^ { 2 } + 3 u _ { n } - \frac { 3 } { 2 } .$$

Part A: Conjecture

1. Calculate the exact values, given as irreducible fractions, of $u _ { 1 }$ and $u _ { 2 }$.
2. Give an approximate value to $10 ^ { - 5 }$ of the terms $u _ { 3 }$ and $u _ { 4 }$.
3. Conjecture the direction of variation and the convergence of the sequence $(u_n)$.

Part B: Validation of conjectures

Consider the numerical sequence $\left( v _ { n } \right)$ defined for every natural number $n$, by: $v _ { n } = u _ { n } - 3$.

1. Show that, for every natural number $n , v _ { n + 1 } = - \frac { 1 } { 2 } v _ { n } ^ { 2 }$.
2. Prove by induction that, for every natural number $n , - 1 \leqslant v _ { n } \leqslant 0$.
3. a. Prove that, for every natural number $n , v _ { n + 1 } - v _ { n } = - v _ { n } \left( \frac { 1 } { 2 } v _ { n } + 1 \right)$. b. Deduce the direction of variation of the sequence $\left( v _ { n } \right)$.
4. Why can we then affirm that the sequence $\left( v _ { n } \right)$ converges?
5. Let $\ell$ denote the limit of the sequence $(v_n)$. It is admitted that $\ell$ belongs to the interval $[ - 1 ; 0 ]$ and satisfies the equality: $\ell = - \frac { 1 } { 2 } \ell ^ { 2 }$. Determine the value of $\ell$.
6. Are the conjectures made in Part A validated?

A city has a bike-sharing network where two stations A and B are located at the top of a hill. It is admitted that no bikes from other stations arrive at stations A and B.
It is observed that for each hour $n$ on average:

• $20 \%$ of the bikes present at hour $n - 1$ at station A are still at this station. $60 \%$ of the bikes present at hour $n - 1$ at station A are at station B and the others are in other stations of the network or in circulation.
• $10 \%$ of the bikes present at hour $n - 1$ at station B are at station $\mathrm { A } , 30 \%$ are still at station B and the others are in other stations of the network or in circulation.
• At the beginning of the day, station A has 50 bikes, station B has 60 bikes.

Part A

After $n$ hours, let $a _ { n }$ denote the average number of bikes present at station A and $b _ { n }$ the average number of bikes present at station B. Let $U _ { n }$ denote the column matrix $\binom { a _ { n } } { b _ { n } }$ and thus $U _ { 0 } = \binom { 50 } { 60 }$.

1. Determine the matrix $M$ such that $U _ { n + 1 } = M \times U _ { n }$.
2. Determine $U _ { 1 }$ and $U _ { 2 }$.
3. After how many hours is there only one bike left in station A?

Part B

The service decides to study the effects of a supply of stations A and B consisting of bringing 30 bikes to station A and 10 bikes to station B after each hour of operation.
After $n$ hours, let $\alpha _ { n }$ denote the average number of bikes present at station A and $\beta _ { n }$ the average number of bikes present at station B. Let $V _ { n }$ denote the column matrix $\binom { \alpha _ { n } } { \beta _ { n } }$ and $V _ { 0 } = \binom { 50 } { 60 }$. Under these conditions $V _ { n + 1 } = M \times V _ { n } + R$ with $R = \binom { 30 } { 10 }$.

1. Let $I$ denote the matrix $\left( \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right)$ and $N$ the matrix $I - M$. a. Let $V$ denote a column matrix with two rows. Show that $V = M \times V + R$ is equivalent to $N \times V = R$. b. It is admitted that $N$ is an invertible matrix and that $N ^ { - 1 } = \left( \begin{array} { l l } 1,4 & 0,2 \\ 1,2 & 1,6 \end{array} \right)$. Deduce that $V = \binom { 44 } { 52 }$.
2. For every natural number $n$, let $W _ { n } = V _ { n } - V$. a. Show that $W _ { n + 1 } = M \times W _ { n }$. b. It is admitted that: for every natural number $n , W _ { n } = M ^ { n } \times W _ { 0 }$, and $$\text{for every natural number } n \geqslant 1 , M ^ { n } = \frac { 1 } { 2 ^ { n - 1 } } \left( \begin{array} { l l } 0,2 & 0,1 \\ 0,6 & 0,3 \end{array} \right) .$$ Calculate, for every natural number $n \geqslant 1 , V _ { n }$ as a function of $n$. c. Does the average number of bikes present in stations A and B tend to stabilize?

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.

Exercise 4 — Candidates who have not chosen the specialization option

Let the numerical sequence ( $u _ { n }$ ) defined on the set of natural integers $\mathbb { N }$ by
$$\left\{ \begin{aligned} u _ { 0 } & = 2 \\ \text { and for all natural integer } n , u _ { n + 1 } & = \frac { 1 } { 5 } u _ { n } + 3 \times 0{,}5 ^ { n } . \end{aligned} \right.$$

1. a. Copy and, using a calculator, complete the table of values of the sequence $\left( u _ { n } \right)$ approximated to $10 ^ { - 2 }$ near:
   

   $n$	0	1	2	3	4	5	6	7	8
   $u _ { n }$	2

   
   b. Based on this table, state a conjecture about the direction of variation of the sequence $\left( u _ { n } \right)$.
   
2. a. Prove, by induction, that for all non-zero natural integer $n$ we have $$u _ { n } \geqslant \frac { 15 } { 4 } \times 0{,}5 ^ { n }$$ b. Deduce that, for all natural integer $n$ non-zero, $u _ { n + 1 } - u _ { n } \leqslant 0$. c. Prove that the sequence ( $u _ { n }$ ) is convergent.
   
3. We propose, in this question, to determine the limit of the sequence $\left( u _ { n } \right)$. Let $\left( v _ { n } \right)$ be the sequence defined on $\mathbb { N }$ by $v _ { n } = u _ { n } - 10 \times 0{,}5 ^ { n }$. a. Prove that the sequence $\left( v _ { n } \right)$ is a geometric sequence with ratio $\frac { 1 } { 5 }$. We will specify the first term of the sequence $\left( v _ { n } \right)$. b. Deduce that for all natural integer $n$, $$u _ { n } = - 8 \times \left( \frac { 1 } { 5 } \right) ^ { n } + 10 \times 0{,}5 ^ { n }.$$ c. Determine the limit of the sequence ( $u _ { n }$ ).
   
4. Copy and complete lines (1), (2) and (3) of the following algorithm, so that it displays the smallest value of $n$ such that $u _ { n } \leqslant 0{,}01$.
   

   Input:	$n$ and $u$ are numbers
   Initialization :	$n$ takes the value 0
   	$u$ takes the value 2
   Processing :	While $\ldots$	(1)
   	$n$ takes the value $\ldots$	(2)
   	$u$ takes the value $\ldots$	(3)
   	End While
   Output:	Display $n$

   
   

Exercise 4 — Candidates who have followed the specialization course
A fish farmer has two basins A and B for raising his fish. Every year at the same time:

• he empties basin B and sells all the fish it contained and transfers all the fish from basin A to basin B;
• the sale of each fish allows the purchase of two small fish intended for basin A.

Furthermore, the fish farmer buys an additional 200 fish for basin A and 100 fish for basin B. For every natural integer greater than or equal to 1, we denote respectively by $a _ { n }$ and $b _ { n }$ the numbers of fish in basins A and B after $n$ years. At the beginning of the first year, the number of fish in basin A is $a _ { 0 } = 200$ and that in basin B is $b _ { 0 } = 100$.

1. Justify that $a _ { 1 } = 400$ and $b _ { 1 } = 300$ then calculate $a _ { 2 }$ and $b _ { 2 }$.
2. We denote by $A$ and $B$ the matrices such that $A = \left( \begin{array} { l l } 0 & 2 \\ 1 & 0 \end{array} \right)$ and $B = \binom { 200 } { 100 }$ and for every natural integer $n$, we set $X _ { n } = \binom { a _ { n } } { b _ { n } }$. a. Explain why for every natural integer $n , X _ { n + 1 } = A X _ { n } + B$. b. Determine the real numbers $x$ and $y$ such that $\binom { x } { y } = A \binom { x } { y } + B$. c. For every natural integer $n$, we set $Y _ { n } = \binom { a _ { n } + 400 } { b _ { n } + 300 }$. Prove that for every natural integer $n , Y _ { n + 1 } = A Y _ { n }$.
3. For every natural integer $n$, we set $Z _ { n } = Y _ { 2 n }$. a. Prove that for every natural integer $n , Z _ { n + 1 } = A ^ { 2 } Z _ { n }$. Deduce that for every natural integer $n , Z _ { n + 1 } = 2 Z _ { n }$. b. We admit that this recurrence relation allows us to conclude that for every natural integer $n$, $$Y _ { 2 n } = 2 ^ { n } Y _ { 0 }$$ Deduce that $Y _ { 2 n + 1 } = 2 ^ { n } Y _ { 1 }$ then prove that for every natural integer $n$, $$a _ { 2 n } = 600 \times 2 ^ { n } - 400 \text { and } a _ { 2 n + 1 } = 800 \times 2 ^ { n } - 400 .$$
4. Basin A has a capacity limited to 10000 fish. a. An algorithm is given that, for a given value of $p$, computes the number of fish $a$ in basin A after $p$ years using the formulas $a_{2n} = 600 \times 2^n - 400$ and $a_{2n+1} = 800 \times 2^n - 400$. Use this algorithm to determine from which year the capacity of basin A is exceeded.

Exercise 2 (5 points) -- Common to all candidates

Part A

Let $(u_n)$ be the sequence defined by its first term $u_0$ and, for every natural number $n$, by the relation $$u_{n+1} = a u_n + b \quad (a \text{ and } b \text{ non-zero real numbers such that } a \neq 1).$$ We set, for every natural number $n$, $\quad v_n = u_n - \dfrac{b}{1-a}$.

1. Prove that the sequence $(v_n)$ is geometric with common ratio $a$.
2. Deduce that if $a$ belongs to the interval $]-1\,;\,1[$, then the sequence $(u_n)$ has limit $\dfrac{b}{1-a}$.

Part B

In March 2015, Max buys a green plant measuring 80 cm. He is advised to prune it every year, in March, by cutting a quarter of its height. The plant will then grow 30 cm over the following twelve months. As soon as he gets home, Max prunes his plant.

1. What will be the height of the plant in March 2016 before Max prunes it?
2. For every natural number $n$, we denote by $h_n$ the height of the plant, before pruning, in March of the year $(2015 + n)$. a. Justify that, for every natural number $n$, $\quad h_{n+1} = 0.75\,h_n + 30$. b. Conjecture using a calculator the direction of variation of the sequence $(h_n)$. Prove this conjecture (you may use a proof by induction). c. Is the sequence $(h_n)$ convergent? Justify your answer.

Candidates who have followed the specialization course
A smoker decides to quit smoking. We choose to use the following model:

• if they do not smoke on a given day, they do not smoke the next day with a probability of 0.9;
• if they smoke on a given day, they smoke the next day with a probability of 0.6.

We call $p_{n}$ the probability of not smoking on the $n$-th day after their decision to quit smoking and $q_{n}$, the probability of smoking on the $n$-th day after their decision to quit smoking. We assume that $p_{0} = 0$ and $q_{0} = 1$.

1. Calculate $p_{1}$ and $q_{1}$.
2. We use a spreadsheet to automate the calculation of successive terms of the sequences $(p_{n})$ and $(q_{n})$. A screenshot of this spreadsheet is provided below:
   

   \cline{2-5} \multicolumn{1}{c|}{}	A	B	C	D
   1	$n$	$p_{n}$	$q_{n}$
   2	0	0	1
   3	1
   4	2
   5	3

   
   Column A contains the values of the natural integer $n$. What formulas can be written in cells B3 and C3 so that by copying them downward, we obtain respectively in columns B and C the successive terms of the sequences $(p_{n})$ and $(q_{n})$?
3. We define the matrices $M$ and, for every natural integer $n$, $X_{n}$ by
   $$M = \left(\begin{array}{ll} 0,9 & 0,4 \\ 0,1 & 0,6 \end{array}\right) \quad \text{and} \quad X_{n} = \binom{p_{n}}{q_{n}}.$$
   We admit that $X_{n+1} = M \times X_{n}$ and that, for every natural integer $n$, $X_n = M^{n} \times X_{0}$. We define the matrices $A$ and $B$ by $A = \left(\begin{array}{ll} 0,8 & 0,8 \\ 0,2 & 0,2 \end{array}\right)$ and $B = \left(\begin{array}{cc} 0,2 & -0,8 \\ -0,2 & 0,8 \end{array}\right)$. a) Prove that $M = A + 0,5 B$. b) Verify that $A^{2} = A$, and that $A \times B = B \times A = \left(\begin{array}{ll} 0 & 0 \\ 0 & 0 \end{array}\right)$.
   We admit in the following that, for every strictly positive natural integer $n$, $A^{n} = A$ and $B^{n} = B$. c) Prove that, for every natural integer $n$, $M^{n} = A + 0,5^{n} B$. d) Deduce that, for every natural integer $n$, $p_{n} = 0,8 - 0,8 \times 0,5^{n}$. e) In the long term, can we assert with certainty that the smoker will quit smoking?