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.

The Maclaurin series for a function $f$ is given by $\sum_{n=1}^{\infty} \frac{(n+1)x^n}{n^2 6^n}$ and converges to $f(x)$ for all $x$ in the interval of convergence. It can be shown that the Maclaurin series for $f$ has a radius of convergence of 6.
(a) Determine whether the Maclaurin series for $f$ converges or diverges at $x = 6$. Give a reason for your answer.
(b) It can be shown that $f(-3) = \sum_{n=1}^{\infty} \frac{(n+1)(-3)^n}{n^2 6^n} = \sum_{n=1}^{\infty} \frac{n+1}{n^2}\left(-\frac{1}{2}\right)^n$ and that the first three terms of this series sum to $S_3 = -\frac{125}{144}$. Show that $\left|f(-3) - S_3\right| < \frac{1}{50}$.
(c) Find the general term of the Maclaurin series for $f'$, the derivative of $f$. Find the radius of convergence of the Maclaurin series for $f'$.
(d) Let $g(x) = \sum_{n=1}^{\infty} \frac{(n+1)x^{2n}}{n^2 3^n}$. Use the ratio test to determine the radius of convergence of the Maclaurin series for $g$.

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 :

Let two sequences $\left( u _ { n } \right)$ and $\left( v _ { n } \right)$ be defined by $u _ { 0 } = 2$ and $v _ { 0 } = 10$ and for every natural number $n$,
$$u _ { n + 1 } = \frac { 2 u _ { n } + v _ { n } } { 3 } \quad \text { and } \quad v _ { n + 1 } = \frac { u _ { n } + 3 v _ { n } } { 4 } .$$
PART A
Consider the following algorithm:
\begin{verbatim} Variables: N is an integer U,V,W are real numbers K is an integer Start: Assign 0 to K Assign 2 to U Assign 10 to V Input N While KExecute this algorithm by inputting $N = 2$. Copy and complete the table given below showing the state of the variables during the execution of the algorithm.

$K$	$W$	$U$	$V$
0
1
2

PART B
1. a. Show that for every natural number $n , v _ { n + 1 } - u _ { n + 1 } = \frac { 5 } { 12 } \left( v _ { n } - u _ { n } \right)$.
b. For every natural number $n$ let $w _ { n } = v _ { n } - u _ { n }$.
Show that for every natural number $n , w _ { n } = 8 \left( \frac { 5 } { 12 } \right) ^ { n }$.
2. a. Prove that the sequence $( u _ { n } )$ is increasing and that the sequence $( v _ { n } )$ is decreasing.
b. Deduce from the results of questions 1. b. and 2. a. that for every natural number $n$ we have $u _ { n } \leqslant 10$ and $v _ { n } \geqslant 2$.
c. Deduce that the sequences $\left( u _ { n } \right)$ and $\left( v _ { n } \right)$ are convergent.
3. Show that the sequences $( u _ { n } )$ and $( v _ { n } )$ have the same limit.
4. Show that the sequence $( t _ { n } )$ defined by $t _ { n } = 3 u _ { n } + 4 v _ { n }$ is constant.
Deduce that the common limit of the sequences $\left( u _ { n } \right)$ and $\left( v _ { n } \right)$ is $\frac { 46 } { 7 }$.

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 $(T _ { n })$.

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 candidates who have NOT followed the specialization course)
We consider the sequence $(u_n)$ defined by $$u_0 = 0 \quad \text{and, for every natural integer } n, u_{n+1} = u_n + 2n + 2.$$

1. Calculate $u_1$ and $u_2$.
2. We consider the following two algorithms:
   

   \multicolumn{2}{|l|}{Algorithm 1}	\multicolumn{2}{|l|}{Algorithm 2}
   Variables :	$n$ is a natural integer $u$ is a real number	Variables :	$n$ is a natural integer $u$ is a real number
   Input : Processing:		Input : Processing:
   	\begin{tabular}{l} Enter the value of $n$ $u$ takes the value 0
   For $i$ ranging from 1 to $n$ : $u$ takes the value $u + 2i + 2$

   & &

   Enter the value of $n$ $u$ takes the value 0

   For $i$ ranging from 0 to $n - 1$ : $u$ takes the value $u + 2i + 2$

   \hline & End For & & End For \hline Output : & Display $u$ & Output : & Display $u$ \hline \end{tabular}
   Of these two algorithms, which one allows the output to display the value of $u_n$, with the value of the natural integer $n$ being entered by the user?
3. Using the algorithm, we obtained the table and the scatter plot below where $n$ is on the horizontal axis and $u_n$ is on the vertical axis.
   

   $n$	$u_n$
   0	0
   1	2
   2	6
   3	12
   4	20
   5	30
   6	42
   7	56
   8	72
   9	90
   10	110
   11	132
   12	156

   
   a. What conjecture can be made about the direction of variation of the sequence $\left( u_n \right)$? Prove this conjecture. b. The parabolic shape of the scatter plot leads to conjecturing the existence of three real numbers $a, b$ and $c$ such that, for every natural integer $n, u_n = an^2 + bn + c$. Within the framework of this conjecture, find the values of $a, b$ and $c$ using the information provided.
4. We define, for every natural integer $n$, the sequence $\left( v_n \right)$ by: $v_n = u_{n+1} - u_n$. a. Express $v_n$ as a function of the natural integer $n$. What is the nature of the sequence $\left( v_n \right)$? b. We define, for every natural integer $n, S_n = \sum_{k=0}^{n} v_k = v_0 + v_1 + \cdots + v_n$. Prove that, for every natural integer $n, S_n = (n+1)(n+2)$. c. Prove that, for every natural integer $n, S_n = u_{n+1} - u_0$, then express $u_n$ as a function of $n$.

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?

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 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.

We denote by $\mathbb{C}$ the set of complex numbers. In the complex plane equipped with an orthonormal coordinate system $(O; \vec{u}, \vec{v})$ we have placed a point $M$ with affixe $z$ belonging to $\mathbb{C}$, then the point $R$ intersection of the circle with center $O$ passing through $M$ and the half-axis $[O; \vec{u})$.
Part A

1. Express the affixe of point $R$ as a function of $z$.
2. Let the point $M'$ with affixe $z'$ defined by $$z' = \frac{1}{2}\left(\frac{z + |z|}{2}\right).$$ Reproduce the figure on the answer sheet and construct the point $M'$.

Part B
We define the sequence of complex numbers $(z_n)$ by a first term $z_0$ belonging to $\mathbb{C}$ and, for every natural integer $n$, by the recurrence relation: $$z_{n+1} = \frac{z_n + |z_n|}{4}$$ The purpose of this part is to study whether the behavior at infinity of the sequence $(|z_n|)$ depends on the choice of $z_0$.

1. What can we say about the behavior at infinity of the sequence $(|z_n|)$ when $z_0$ is a negative real number?
2. What can we say about the behavior at infinity of the sequence $(|z_n|)$ when $z_0$ is a positive real number?
3. We now assume that $z_0$ is not a real number. a. What conjecture can we make about the behavior at infinity of the sequence $(|z_n|)$? b. Prove this conjecture, then conclude.

Let $a$ be a fixed non-zero real number. The purpose of this exercise is to study the sequence $(u_n)$ defined by: $$u_0 = a \quad \text{and, for all } n \text{ in } \mathbb{N}, \quad u_{n+1} = \mathrm{e}^{2u_n} - \mathrm{e}^{u_n}.$$ Note that this equality can also be written: $u_{n+1} = e^{u_n}(\mathrm{e}^{u_n} - 1)$.

1. Let $g$ be the function defined for all real $x$ by $$g(x) = \mathrm{e}^{2x} - \mathrm{e}^x - x.$$ a) Calculate $g'(x)$ and prove that, for all real $x$: $g'(x) = (\mathrm{e}^x - 1)(2\mathrm{e}^x + 1)$. b) Determine the variations of the function $g$ and give the value of its minimum. c) By noting that $u_{n+1} - u_n = g(u_n)$, study the direction of variation of the sequence $(u_n)$.
   
2. In this question, we assume that $a \leqslant 0$. a) Prove by induction that, for all natural integer $n$, $u_n \leqslant 0$. b) Deduce from the previous questions that the sequence $(u_n)$ is convergent. c) In the case where $a$ equals 0, give the limit of the sequence $(u_n)$.
   
3. In this question, we assume that $a > 0$.
   Since the sequence $(u_n)$ is increasing, question 1 allows us to assert that, for all natural integer $n$, $u_n \geqslant a$. a) Prove that, for all natural integer $n$, we have: $u_{n+1} - u_n \geqslant g(a)$. b) Prove by induction that, for all natural integer $n$, we have: $$u_n \geqslant a + n \times g(a).$$ c) Determine the limit of the sequence $(u_n)$.
   
4. In this question, we take $a = 0.02$.
   The following algorithm is intended to determine the smallest integer $n$ such that $u_n > M$, where $M$ denotes a positive real number. This algorithm is incomplete.
   

   Variables	$n$ is an integer, $u$ and $M$ are two real numbers
   Initialization	\begin{tabular}{l} $u$ takes the value 0.02
   $n$ takes the value 0
   Enter the value of $M$

   \hline Processing & While $\cdots$ & $\ldots$ & $\ldots$ End while & \end{tabular}
   a) On your paper, rewrite the ``Processing'' part by completing it. b) Using a calculator, determine the value that this algorithm will display if $M = 60$.

In a country with a constant population equal to 120 million, the inhabitants live either in rural areas or in cities. Population movements can be modeled as follows:

• in 2010, the population consists of 90 million rural inhabitants and 30 million city dwellers;
• each year, $10\%$ of rural inhabitants migrate to the city;
• each year, $5\%$ of city dwellers migrate to rural areas.

For any natural integer $n$, we denote:

• $u_n$ the population in rural areas, in the year $2010 + n$, expressed in millions of inhabitants;
• $v_n$ the population in cities, in the year $2010 + n$, expressed in millions of inhabitants.

We have $u_0 = 90$ and $v_0 = 30$.
Part A

1. Translate the fact that the total population is constant by a relation linking $u_n$ and $v_n$.
2. What formulas can be entered in cells B3 and C3 which, copied downward, allow us to obtain the spreadsheet showing the evolution of $(u_n)$ and $(v_n)$?
3. What conjectures can be made concerning the long-term evolution of this population?

Part B
Deduce from the recurrence relations that, for every natural integer $n$, $R_n = 50 \times 0.85^n + 40$ and determine the expression of $C_n$ as a function of $n$. b. Determine the limit of $R_n$ and of $C_n$ when $n$ tends towards $+\infty$. What can we conclude from this for the population studied? 6. a. Complete the algorithm so that it displays the number of years after which the urban population will exceed the rural population. b. By solving the inequality with unknown $n$, $$50 \times 0.85^n + 40 < 80 - 50 \times 0.85^n,$$ find again the value displayed by the algorithm.

(Candidates who have not followed the specialization course)
Part A
We consider the following algorithm:

\begin{tabular}{l} Variables:

Input: Processing:

Output:

&

$k$ and $p$ are natural integers

$u$ is a real number

Ask for the value of $p$

Assign to $u$ the value 5

For $k$ varying from 1 to $p$

Assign to $u$ the value $0,5u + 0,5(k-1) - 1,5$

End for

Display $u$

\hline \end{tabular}
Run this algorithm for $p = 2$ by indicating the values of the variables at each step. What number do we obtain as output?
Part B
Let $(u_n)$ be the sequence defined by its first term $u_0 = 5$ and, for every natural integer $n$ by $$u_{n+1} = 0,5u_n + 0,5n - 1,5.$$

1. Modify the algorithm from the first part to obtain as output all the values of $u_n$ for $n$ varying from 1 to $p$.
2. Using the modified algorithm, after entering $p = 4$, we obtain the following results:
   

   $n$	1	2	3	4
   $u_n$	1	$-0,5$	$-0,75$	$-0,375$

   
   Can we assert, based on these results, that the sequence $(u_n)$ is decreasing? Justify.
3. Prove by induction that for every natural integer $n$ greater than or equal to 3, $u_{n+1} > u_n$. What can we deduce about the monotonicity of the sequence $(u_n)$?
4. Let $(v_n)$ be the sequence defined for every natural integer $n$ by $v_n = 0,1u_n - 0,1n + 0,5$. Prove that the sequence $(v_n)$ is geometric with ratio 0,5 and express $v_n$ as a function of $n$.
5. Deduce that, for every natural integer $n$, $u_n = 2^{1-n} \cdot 5 + n - 5$ (or the equivalent closed form expression for $u_n$ as a function of $n$).

(For candidates who have not followed the specialization course)
We consider two sequences of real numbers $( d _ { n } )$ and $( a _ { n } )$ defined by $d _ { 0 } = 300$, $a _ { 0 } = 450$ and, for every natural number $n \geqslant 0$
$$\left\{ \begin{array} { l } d _ { n + 1 } = \frac { 1 } { 2 } d _ { n } + 100 \\ a _ { n + 1 } = \frac { 1 } { 2 } d _ { n } + \frac { 1 } { 2 } a _ { n } + 70 \end{array} \right.$$

1. Calculate $d _ { 1 }$ and $a _ { 1 }$.
2. It is desired to write an algorithm that allows displaying as output the values of $d _ { n }$ and $a _ { n }$ for an integer value of $n$ entered by the user. The following algorithm is proposed:
   

   Variables:	\begin{tabular}{l} $n$ and $k$ are natural numbers
   $D$ and $A$ are real numbers

   \hline Initialization: &

   $D$ takes the value 300

   $A$ takes the value 450

   Enter the value of $n$

   \hline Processing: &

   For $k$ varying from 1 to $n$

   $D$ takes the value $\frac { D } { 2 } + 100$

   $A$ takes the value $\frac { A } { 2 } + \frac { D } { 2 } + 70$

   End for

   \hline Output: &

   Display $D$

   Display $A$

   \hline \end{tabular}
   a. What numbers are obtained as output of the algorithm for $n = 1$? Are these results consistent with those obtained in question 1? b. Explain how to correct this algorithm so that it displays the desired results.
3. a. For every natural number $n$, we set $e _ { n } = d _ { n } - 200$. Show that the sequence $( e _ { n } )$ is geometric. b. Deduce the expression of $d _ { n }$ as a function of $n$. c. Is the sequence $( d _ { n } )$ convergent? Justify.
4. We admit that for every natural number $n$,
   $$a _ { n } = 100 n \left( \frac { 1 } { 2 } \right) ^ { n } + 110 \left( \frac { 1 } { 2 } \right) ^ { n } + 340 .$$
   a. Show that for every integer $n$ greater than or equal to 3, we have $2 n ^ { 2 } \geqslant ( n + 1 ) ^ { 2 }$. b. Show by induction that for every integer $n$ greater than or equal to 4, $2 ^ { n } \geqslant n ^ { 2 }$. c. Deduce that for every integer $n$ greater than or equal to 4,
   $$0 \leqslant 100 n \left( \frac { 1 } { 2 } \right) ^ { n } \leqslant \frac { 100 } { n } .$$
   d. Study the convergence of the sequence $\left( a _ { n } \right)$.

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 $\left( S _ { n } \right)$.

Exercise 2

Let $u$ be the sequence defined by $u _ { 0 } = 2$ and, for every natural integer $n$, by $$u _ { n + 1 } = 2 u _ { n } + 2 n ^ { 2 } - n .$$ We also consider the sequence $v$ defined, for every natural integer $n$, by $$v _ { n } = u _ { n } + 2 n ^ { 2 } + 3 n + 5$$

1. Here is an extract from a spreadsheet:
   

   	A	B	C
   1	$n$	$u$	$v$
   2	0	2	7
   3	1	4	14
   4	2	9	28
   5	3	24	56
   6	4	63
   7

   
   What formulas were written in cells C2 and B3 and copied downward to display the terms of the sequences $u$ and $v$?
2. Determine, by justifying, an expression of $v _ { n }$ and of $u _ { n }$ as a function of $n$ only.

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 $\ell$.

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 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 $u_r \geqslant 50$.