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 $( u _ { n } )$.

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?

Exercise 4 — For candidates who have NOT followed the speciality course We consider the function $f$ defined on $\mathbb{R}$ by: $$f(x) = \frac{1}{2}x^{2} - x + \frac{3}{2}$$ Let $a$ be a positive real number. We define the sequence $(u_{n})$ by $u_{0} = a$ and, for every natural number $n$: $u_{n+1} = f(u_{n})$. The purpose of this exercise is to study the behaviour of the sequence $(u_{n})$ as $n$ tends to $+\infty$, depending on different values of its first term $u_{0} = a$.

1. Using a calculator, conjecture the behaviour of the sequence $(u_{n})$ as $n$ tends to $+\infty$, for $a = 2.9$ then for $a = 3.1$.
2. In this question, we assume that the sequence $(u_{n})$ converges to a real number $\ell$. a. By noting that $u_{n+1} = \frac{1}{2}u_{n}^{2} - u_{n} + \frac{3}{2}$, show that $\ell = \frac{1}{2}\ell^{2} - \ell + \frac{3}{2}$. b. Show that the possible values of $\ell$ are 1 and 3.
3. In this question, we take $a = 2.9$. a. Show that $f$ is increasing on the interval $[1; +\infty[$. b. Show by induction that, for every natural number $n$, we have: $1 \leqslant u_{n+1} \leqslant u_{n}$. c. Show that $(u_{n})$ converges and determine its limit.
4. In this question, we take $a = 3.1$ and we admit that the sequence $(u_{n})$ is increasing. a. Using the previous questions show that the sequence $(u_{n})$ is not bounded above. b. Deduce the behaviour of the sequence $(u_{n})$ as $n$ tends to $+\infty$. c. The following algorithm calculates the smallest rank $p$ for which $u_{p} > 10^{6}$. Copy and complete this algorithm. $P$ is a natural number and $U$ is a real number. \begin{verbatim} P <- 0 U..... Tant que... P ...... U ...... Fin Tant que \end{verbatim}

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

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 $10 ^ { - 15 }$.

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 all natural number $n$, $u_{n+1} = u_n - \ln(1 + u_n)$. 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 all natural number $n$, $u_n \geqslant 0$. b. Prove that the sequence $(u_n)$ is decreasing, and deduce that for all natural number $n$, $u_n \leqslant 1$. 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.

Exercise 4 (5 points) — Candidates who have followed the specialization course
In a public garden, an artist must install an aquatic artwork. This artwork will consist of two basins A and B as well as a filtering reserve R. Initially, the two basins each contain 100 liters of water. A system of pipes allows the following water transfers to be carried out, every hour and in this order:

• first, half of basin A empties into reserve R;
• then, three quarters of basin B empty into basin A;
• finally, 200 liters of water are added to basin A and 300 liters of water are added to basin B.

The quantities of water in the two basins A and B are modeled using two sequences $(a_n)$ and $(b_n)$: for any natural number $n$, we denote by $a_n$ and $b_n$ the quantities of water in hundreds of liters that will be respectively contained in basins A and B after $n$ hours. For any natural number $n$, we denote by $U_n$ the column matrix $U_n = \binom{a_n}{b_n}$. Thus $U_0 = \binom{1}{1}$.
1. Justify that, for any natural number $n$, $U_{n+1} = MU_n + C$ where $M = \left(\begin{array}{cc} 0.5 & 0.75 \\ 0 & 0.25 \end{array}\right)$ and $C = \binom{2}{3}$.
2. Consider the matrix $P = \left(\begin{array}{cc} 1 & 3 \\ 0 & -1 \end{array}\right)$.
a. Calculate $P^2$. Deduce that the matrix $P$ is invertible and specify its inverse matrix.
b. Show that $PMP$ is a diagonal matrix $D$ that you will specify.
c. Calculate $PDP$.
d. Prove by induction that, for any natural number $n$, $M^n = PD^nP$.
It is admitted henceforth that for any natural number $n$, $M^n = \left(\begin{array}{cc} 0.5^n & 3 \times 0.5^n - 3 \times 0.25^n \\ 0 & 0.25^n \end{array}\right)$.
3. Show that the matrix $X = \binom{10}{4}$ satisfies $X = MX + C$.
4. For any natural number $n$, we define the matrix $V_n$ by $V_n = U_n - X$.
a. Show that for any natural number $n$, $V_{n+1} = MV_n$.
b. It is admitted that, for any non-zero natural number $n$, $V_n = M^n V_0$. Show that for any non-zero natural number $n$, $$U_n = \binom{-18 \times 0.5^n + 9 \times 0.25^n + 10}{-3 \times 0.25^n + 4}.$$
5. a. Show that the sequence $(b_n)$ is increasing and bounded above. Determine its limit.
b. Determine the limit of the sequence $(a_n)$.
c. It is admitted that the sequence $(a_n)$ is increasing. Deduce the capacity of the two basins A and B that must be planned for the feasibility of the project, that is, to avoid any overflow.

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

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

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

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

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

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

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

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

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

A biologist is interested in the evolution of the population of an animal species on an island in the Pacific. At the beginning of 2020, this population had 600 individuals. We consider that the species will be threatened with extinction on this island if its population becomes less than or equal to 20 individuals. The biologist models the number of individuals by the sequence $(u_n)$ defined by:
$$\begin{cases} u_{0} & = 0.6 \\ u_{n+1} & = 0.75 u_{n} \left( 1 - 0.15 u_{n} \right) \end{cases}$$
where for every natural integer $n$, $u_{n}$ denotes the number of individuals, in thousands, at the beginning of the year $2020 + n$.

1. Estimate, according to this model, the number of individuals present on the island at the beginning of 2021 and then at the beginning of 2022.

Let $f$ be the function defined on the interval $[ 0 ; 1 ]$ by
$$f ( x ) = 0.75 x ( 1 - 0.15 x )$$

1. Show that the function $f$ is increasing on the interval $[ 0 ; 1 ]$ and draw up its variation table.
2. Solve in the interval $[ 0 ; 1 ]$ the equation $f ( x ) = x$.

We note for the rest of the exercise that, for every natural integer $n$, $u_{n+1} = f \left( u_{n} \right)$.
4. a. Prove by induction that for every natural integer $n$, $0 \leqslant u_{n+1} \leqslant u_{n} \leqslant 1$. b. Deduce that the sequence $\left( u_{n} \right)$ is convergent. c. Determine the limit $\ell$ of the sequence $(u_{n})$.
5. The biologist has the intuition that the species will sooner or later be threatened with extinction. a. Justify that, according to this model, the biologist is correct. b. The biologist has programmed in Python language the function menace() below:
\begin{verbatim} def menace() : u = 0.6 n = 0 while u>0.02 : u=0.75*u*(1-0.15*u) n = n+1 return n \end{verbatim}
Give the numerical value returned when the function menace() is called. Interpret this result in the context of the exercise.

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

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

\hline \end{tabular}

We consider the sequences $(u_{n})$ and $(v_{n})$ defined for every natural integer $n$ by:
$$\left\{ \begin{array}{l} u_{0} = v_{0} = 1 \\ u_{n+1} = u_{n} + v_{n} \\ v_{n+1} = 2u_{n} + v_{n} \end{array} \right.$$
Throughout the rest of the exercise, we assume that the sequences $(u_{n})$ and $(v_{n})$ are strictly positive.

1. a. Calculate $u_{1}$ and $v_{1}$. b. Prove that the sequence $(v_{n})$ is strictly increasing, then deduce that for every natural integer $n$, $v_{n} \geqslant 1$. c. Prove by induction that for every natural integer $n$, we have: $u_{n} \geqslant n + 1$. d. Deduce the limit of the sequence $(u_{n})$.
2. We set, for every natural integer $n$: $$r_{n} = \frac{v_{n}}{u_{n}}.$$ We assume that: $$r_{n}^{2} = 2 + \frac{(-1)^{n+1}}{u_{n}^{2}}$$ a. Prove that for every natural integer $n$: $$-\frac{1}{u_{n}^{2}} \leqslant \frac{(-1)^{n+1}}{u_{n}^{2}} \leqslant \frac{1}{u_{n}^{2}}.$$ b. Deduce: $$\lim_{n \rightarrow +\infty} \frac{(-1)^{n+1}}{u_{n}^{2}}$$ c. Determine the limit of the sequence $\left(r_{n}^{2}\right)$ and deduce that $\left(r_{n}\right)$ converges to $\sqrt{2}$. d. Prove that for every natural integer $n$, $$r_{n+1} = \frac{2 + r_{n}}{1 + r_{n}}$$ e. Consider the following program written in Python language: \begin{verbatim} def seuil() : n = 0 r = l while abs(r-sqrt(2)) > 10**(-4) : r = (2+r)/(1+r) n = n+1 return n \end{verbatim} (abs denotes absolute value, sqrt the square root and $10^{**}(-4)$ represents $10^{-4}$). The value of $n$ returned by this program is 5. What does it correspond to?

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

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

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

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

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

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

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

Part A

Consider the function $f$ defined on the interval $[ 1 ; + \infty [$ by $$f ( x ) = \frac { \ln x } { x }$$ where ln denotes the natural logarithm function.

1. Give the limit of the function $f$ at $+ \infty$.
2. We admit that the function $f$ is differentiable on the interval $[ 1 ; + \infty [$ and we denote by $f ^ { \prime }$ its derivative function. a. Show that, for every real number $x \geqslant 1$, $f ^ { \prime } ( x ) = \frac { 1 - \ln x } { x ^ { 2 } }$. b. Justify the following sign table, giving the sign of $f ^ { \prime } ( x )$ according to the values of $x$.
   

   $x$	1		e	$+ \infty$
   $f ^ { \prime } ( x )$		+	0	-

   
   c. Draw up the complete variation table of the function $f$.
3. Let $k$ be a non-negative real number. a. Show that, if $0 \leqslant k \leqslant \frac { 1 } { \mathrm { e } }$, the equation $f ( x ) = k$ admits a unique solution on the interval $[1; e]$. b. If $k > \frac { 1 } { \mathrm { e } }$, does the equation $f ( x ) = k$ admit solutions on the interval $[ 1 ; + \infty [$? Justify.

Part B

Let $g$ be the function defined on $\mathbb { R }$ by: $$g ( x ) = \mathrm { e } ^ { \frac { x } { 4 } } .$$ We consider the sequence $\left( u _ { n } \right)$ defined by $u _ { 0 } = 1$ and, for every natural integer $n$: $u _ { n + 1 } = e ^ { \frac { u _ { n } } { 4 } }$, that is: $u _ { n + 1 } = g \left( u _ { n } \right)$.

1. Justify that the function $g$ is increasing on $\mathbb { R }$.
2. Show by induction that, for every natural integer $n$, we have: $u _ { n } \leqslant u _ { n + 1 } \leqslant \mathrm { e }$.
3. Deduce that the sequence $( u _ { n } )$ is convergent.

We denote by $\ell$ the limit of the sequence $( u _ { n } )$ and we admit that $\ell$ is a solution of the equation: $$\mathrm { e } ^ { \frac { x } { 4 } } = x .$$

1. Deduce that $\ell$ is a solution of the equation $f ( x ) = \frac { 1 } { 4 }$, where $f$ is the function studied in Part A.
2. Give an approximate value to $10 ^ { - 2 }$ near of the limit $\ell$ of the sequence $( u _ { n } )$.

Exercise 2 — 7 points

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

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

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

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

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

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

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

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

Exercise 3 — Theme: Functions; Sequences This exercise is a multiple choice questionnaire. For each of the following questions, only one of the four proposed answers is correct. A wrong answer, multiple answers, or no answer to a question earns no points and loses no points. To answer, indicate on your paper the question number and the letter of the chosen answer. No justification is required.

1. Let $g$ be the function defined on $\mathbb{R}$ by $g(x) = x^{1000} + x$. We can affirm that: a. the function $g$ is concave on $\mathbb{R}$. b. the function $g$ is convex on $\mathbb{R}$. c. the function $g$ has exactly one inflection point. d. the function $g$ has exactly two inflection points.
2. Consider a function $f$ defined and differentiable on $\mathbb{R}$. Let $f'$ denote its derivative function. Let $\mathscr{C}$ denote the representative curve of $f$. Let $\Gamma$ denote the representative curve of $f'$. The curve $\Gamma$ is plotted below. Let $T$ denote the tangent to the curve $\mathscr{C}$ at the point with abscissa 0. We can affirm that the tangent $T$ is parallel to the line with equation: a. $y = x$ b. $y = 0$ c. $y = 1$ d. $x = 0$
3. Consider the sequence $(u_n)$ defined for every natural number $n$ by $u_n = \frac{(-1)^n}{n+1}$. We can affirm that the sequence $(u_n)$ is: a. bounded above and not bounded below. b. bounded below and not bounded above. c. bounded. d. not bounded above and not bounded below.
4. Let $k$ be a non-zero real number. Let $(v_n)$ be a sequence defined for every natural number $n$. Suppose that $v_0 = k$ and that for all $n$, we have $v_n \times v_{n+1} < 0$. We can affirm that $v_{10}$ is: a. positive. b. negative. c. of the same sign as $k$. d. of the same sign as $-k$.
5. Consider the sequence $(w_n)$ defined for every natural number $n$ by: $$w_{n+1} = 2w_n - 4 \quad \text{and} \quad w_2 = 8.$$ We can affirm that: a. $w_0 = 0$ b. $w_0 = 5$. c. $w_0 = 10$. d. It is not possible to calculate $w_0$.
6. Consider the sequence $(a_n)$ defined for every natural number $n$ by: $$a_{n+1} = \frac{\mathrm{e}^n}{\mathrm{e}^n + 1} a_n \quad \text{and} \quad a_0 = 1.$$ We can affirm that: a. the sequence $(a_n)$ is strictly increasing. b. the sequence $(a_n)$ is strictly decreasing. c. the sequence $(a_n)$ is not monotone. d. the sequence $(a_n)$ is constant.
7. A cell reproduces by dividing into two identical cells, which divide in turn, and so on. The generation time is defined as the time required for a given cell to divide into two cells. 1 cell was placed in culture. After 4 hours, there are approximately 4000 cells. We can affirm that the generation time is approximately equal to: a. less than one minute. b. 12 minutes. c. 20 minutes. d. 1 hour.

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

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

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

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

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

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

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

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