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

We consider the sequence $\left( u _ { n } \right) _ { n \in \mathbb { N } }$ defined by $u _ { 0 } = 400$ and for every natural integer $n$:
$$u _ { n + 1 } = 0,9 u _ { n } + 60 .$$

1. a. Calculate $u _ { 1 }$ and $u _ { 2 }$. b. Conjecture the direction of variation of the sequence $\left( u _ { n } \right) _ { n \in \mathbb { N } }$
2. Show, by induction, that for every natural integer $n$, we have the inequality $$0 \leqslant u _ { n } \leqslant u _ { n + 1 } \leqslant 600$$
3. a. Show that the sequence $\left( u _ { n } \right) _ { n \in \mathbb { N } }$ is convergent. b. Determine the limit of the sequence $\left( u _ { n } \right) _ { n \in \mathbb { N } }$. Justify.
4. A function is given written in Python language: \begin{verbatim} def mystere(seuil) : n=0 u=400 while u <= seuil : n = n+1 u=0.9*u+60 return n \end{verbatim} What value do we obtain by typing in the Python console: mystere(500)?

Part B: A fruit grower owns an orchard where he has room to grow a maximum of 500 trees. Each year he sells $10\%$ of the trees in his orchard and then he plants 60 new trees. The orchard has 400 trees in 2023. The fruit grower thinks he will be able to continue selling and planting trees at the same rate in the coming years. Will he face a space problem in his orchard? Explain your answer.

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

Part A

Consider the function $f$ defined by :
$$f ( x ) = x - \ln ( 1 + x ) .$$

1. Justify that the function $f$ is defined on the interval $] - 1 ; + \infty [$.
2. We admit that the function $f$ is differentiable on $] - 1 ; + \infty [$.

Determine the expression of its derivative function $f ^ { \prime } ( x )$.
3. a. Deduce the direction of variation of the function $f$ on the interval $] - 1 ; + \infty [$. b. Deduce the sign of the function $f$ on the interval $] - 1 ; 0 [$.
4. a. Show that, for all $x$ belonging to the interval $] - 1 ; + \infty [$, we have :
$$f ( x ) = \ln \left( \frac { \mathrm { e } ^ { x } } { 1 + x } \right) .$$
b. Deduce the limit at $+ \infty$ of the function $f$.

Part B

Consider the sequence ( $u _ { n }$ ) defined by $u _ { 0 } = 10$ and, for all natural integer $n$,
$$u _ { n + 1 } = u _ { n } - \ln \left( 1 + u _ { n } \right) .$$
We admit that the sequence ( $u _ { n }$ ) is well defined.

1. Give the value rounded to the nearest thousandth of $u _ { 1 }$.
2. Using question 3. a. of Part A , prove by induction that, for all natural integer $n$, we have $u _ { n } \geqslant 0$.
3. Prove that the sequence ( $u _ { n }$ ) is decreasing.
4. Deduce from the previous questions that the sequence ( $u _ { n }$ ) converges.
5. Determine the limit of the sequence $\left( u _ { n } \right)$.

We consider the sequence $(u_n)$ defined by $u_0 = 8$ and, for every natural number $n$, $$u_{n+1} = \frac{6u_n + 2}{u_n + 5}.$$

1. Calculate $u_1$.
2. Let $f$ be the function defined on the interval $[0; +\infty[$ by: $$f(x) = \frac{6x + 2}{x + 5}.$$ Thus, for every natural number $n$, we have: $u_{n+1} = f(u_n)$.
   a. Prove that the function $f$ is strictly increasing on the interval $[0; +\infty[$. Deduce that for every real number $x > 2$, we have $f(x) > 2$.
   b. Prove by induction that, for every natural number $n$, we have $u_n > 2$.
   
3. We admit that, for every natural number $n$, we have: $$u_{n+1} - u_n = \frac{(2 - u_n)(u_n + 1)}{u_n + 5}.$$
   a. Prove that the sequence $(u_n)$ is decreasing.
   b. Deduce that the sequence $(u_n)$ is convergent.
   
4. We define the sequence $(v_n)$ for every natural number by: $$v_n = \frac{u_n - 2}{u_n + 1}.$$
   a. Calculate $v_0$.
   b. Prove that $(v_n)$ is a geometric sequence with common ratio $\dfrac{4}{7}$.
   c. Determine, by justifying, the limit of $(v_n)$. Deduce the limit of $(u_n)$.
   
5. We consider the Python function threshold below, where A is a real number strictly greater than 2. \begin{verbatim} def seuil(A): n = 0 u = 8 while u > A: u = (6*u + 2)/(u + 5) n = n + 1 return n \end{verbatim} Give, without justification, the value returned by the command \texttt{seuil(2.001)} then interpret this value in the context of the exercise.

We consider the sequence ( $u _ { n }$ ) such that $u _ { 0 } = 0$ and for all natural integer $n$ : $$u _ { n + 1 } = \frac { - u _ { n } - 4 } { u _ { n } + 3 } .$$ We admit that $u _ { n }$ is defined for all natural integer $n$.

1. Calculate the exact values of $u _ { 1 }$ and $u _ { 2 }$.
2. We consider the function term below written incompletely in Python language:

\begin{verbatim} def terme (n) : u = ... for i in range(n): u = ... return(u) \end{verbatim}
We recall that in Python language, «i in range (n) » means that $i$ varies from 0 to $n - 1$.
Rewrite and complete the box above so that, for all natural integer $n$, the instruction terme (n) returns the value of $u _ { n }$.
3. Let the function $f$ defined on $] - 3 ; + \infty [$ by: $$f ( x ) = \frac { - x - 4 } { x + 3 }$$ Thus, for all natural integer $n$, we have $u _ { n + 1 } = f \left( u _ { n } \right)$. Prove that the function $f$ is strictly increasing on $] - 3 ; + \infty [$.
4. Prove by induction that for all natural integer $n$ : $$- 2 < u _ { n + 1 } \leqslant u _ { n } .$$

1. Deduce that the sequence ( $u _ { n }$ ) is convergent.
2. Let the sequence $\left( v _ { n } \right)$ defined for all natural integer $n$ by:

$$v _ { n } = \frac { 1 } { u _ { n } + 2 }$$ a. Give $v _ { 0 }$. b. Prove that the sequence ( $v _ { n }$ ) is arithmetic with common difference 1 . c. Deduce that for all natural integer $n \geqslant 1$ : $$u _ { n } = \frac { 1 } { n + 0,5 } - 2 .$$ d. Determine the limit of the sequence $\left( u _ { n } \right)$.

For questions 3. and 4., consider the sequence $(u_n)$ defined on $\mathbb{N}$ by: $$u_0 = 15 \text{ and for every natural number } n : u_{n+1} = 1{,}2\, u_n + 12.$$
The following Python function, whose line 4 is incomplete, must return the smallest value of the integer $n$ such that $u_n > 10000$. \begin{verbatim} def seuil() : n=0 u=15 while ......: n=n+1 u=1,2*u+12 return(n) \end{verbatim} On line 4, we complete with: a. $\mathrm{u} \leqslant 10000$; b. $\mathrm{u} = 10000$ c. $\mathrm{u} > 10000$; d. $n \leqslant 10000$.

Consider the function $f$ defined on $\mathbb{R}$ by $$f(x) = \frac{3}{4}x^2 - 2x + 3$$

1. Draw the table of variations of $f$ on $\mathbb{R}$.
2. Deduce that for all $x$ belonging to the interval $\left[\frac{4}{3}; 2\right]$, $f(x)$ belongs to the interval $\left[\frac{4}{3}; 2\right]$.
3. Prove that for all real $x$, $x \leq f(x)$. For this, one may prove that for all real $x$: $$f(x) - x = \frac{3}{4}(x - 2)^2.$$

Consider the sequence $(u_n)$ defined by a real $u_0$ and for all natural integer $n$: $$u_{n+1} = f(u_n).$$ We have therefore, for all natural integer $n$, $$u_{n+1} = \frac{3}{4}u_n^2 - 2u_n + 3.$$

1. Study of the case: $\frac{4}{3} \leq u_0 \leq 2$. a. Prove by induction that, for all natural integer $n$, $$u_n \leq u_{n+1} \leq 2.$$ b. Deduce that the sequence $(u_n)$ is convergent. c. Prove that the limit of the sequence is equal to 2.
2. Study of the particular case: $u_0 = 3$. It is admitted that in this case the sequence $(u_n)$ tends to $+\infty$. Copy and complete the following ``threshold'' function written in Python, so that it returns the smallest value of $n$ such that $u_n$ is greater than or equal to 100. \begin{verbatim} def seuil() : u = 3 n = 0 while ... u = ... n = ... return n \end{verbatim}
3. Study of the case: $u_0 > 2$. Using a proof by contradiction, show that $(u_n)$ is not convergent.

We consider the sequence $( u _ { n } )$ defined by $u _ { 0 } = 3$ and, for every natural number $n$,
$$u _ { n + 1 } = \frac { 1 } { 2 } u _ { n } + \frac { 1 } { 2 } n + 1 .$$

Part A

This part is a multiple choice questionnaire. For each of the following questions, only one of the four proposed answers is correct. To answer, indicate on your paper the question number and the letter of the chosen answer. No justification is required.
A wrong answer, no answer, or multiple answers, neither earn nor lose points.

1. The value of $u _ { 2 }$ is equal to: a. $\frac { 11 } { 4 }$ b. $\frac { 13 } { 2 }$ b. 2.7 c. 3.5
2. The sequence $\left( v _ { n } \right)$ defined, for every natural number $n$, by $v _ { n } = u _ { n } - n$ is: a. arithmetic with common difference $\frac { 1 } { 2 }$ b. geometric with common ratio $\frac { 1 } { 2 }$ c. constant. d. neither arithmetic nor geometric.
3. We consider the function below, written incompletely in Python language. $n$ denotes a non-zero natural number. We recall that in Python language ``i in range (n)'' means that i varies from 0 to $n - 1$.
   

   1	def terme $( \mathrm { n } )$
   2	$\mathrm { U } = 3$
   3	for i in range(n) :
   4	$\ldots \ldots \ldots \ldots \ldots \ldots \ldots$
   5	return U

   
   For terme(n) to return the value of $u _ { n }$, we can complete line 4 by: a. $\mathrm { U } = \mathrm { U } / 2 + ( \mathrm { i } + 1 ) / 2 + 1$ b. $\mathrm { U } = \mathrm { U } / 2 + \mathrm { n } / 2 + 1$ c. $U = U / 2 + ( i - 1 ) / 2 + 1$ d. $\mathrm { U } = \mathrm { U } / 2 + \mathrm { i } / 2 + 1$

Part B

1. Prove by induction that for every natural number $n$: $$n \leqslant u _ { n } \leqslant n + 3 .$$
2. Deduce the limit of the sequence $( u _ { n } )$.
3. Determine the limit of the sequence $\left( \frac { u _ { n } } { n } \right)$.

We consider the sequence $(u_n)$ defined by $u_0 = 5$ and for every natural number $n$, $$u_{n+1} = \frac{1}{2}\left(u_n + \frac{11}{u_n}\right)$$ We admit that the sequence $(u_n)$ is well defined.
Part A - Study of sequence $(u_n)$

1. Give $u_1$ and $u_2$ in the form of irreducible fractions.
2. We consider the function $f$ defined on the interval $]0; +\infty[$ by: $$f(x) = \frac{1}{2}\left(x + \frac{11}{x}\right)$$ Prove that function $f$ is increasing on the interval $[\sqrt{11}; +\infty[$.
3. Prove by induction that for every natural number $n$, we have: $u_n \geqslant u_{n+1} \geqslant \sqrt{11}$.
4. Deduce that the sequence $(u_n)$ converges to a real limit. We denote this limit by $a$.
5. After determining and solving an equation of which $a$ is a solution, specify the exact value of $a$.

Part B - Geometric application
For every natural number $n$, we consider a rectangle $R_n$ with area 11 whose width is denoted $\ell_n$ and length $L_n$. The sequence $(L_n)$ is defined by $L_0 = 5$ and, for every natural number $n$, $$L_{n+1} = \frac{L_n + \ell_n}{2}$$

1. a. Explain why $\ell_0 = 2.2$. b. Establish that for every natural number $n$, $$\ell_n = \frac{11}{L_n}.$$
2. Verify that the sequence $(L_n)$ corresponds to the sequence $(u_n)$ from Part A.
3. Show that for every natural number $n$, we have $\ell_n \leqslant \sqrt{11} \leqslant L_n$.
4. We admit that the sequences $(L_n)$ and $(\ell_n)$ both converge to $\sqrt{11}$. Interpret this result geometrically in the context of Part B.
5. Here is a script, written in Python language, relating to the sequences studied in this part: \begin{verbatim} def heron(n): L=5 l=2.2 for i in range(n): L = (L+l) / 2 l = 11 / L return round(l, 6), round(L, 6) \end{verbatim} We recall that the Python function round$(\mathrm{x}, \mathrm{k})$ returns a rounded version of the number x with k decimal places. a. If the user types heron(3) in a Python execution console, what output values does he obtain for $\ell$ and $L$? b. Give an interpretation of these two values.

The purpose of Part A is to study the behavior of the sequence $\left( u _ { n } \right)$ defined by $u _ { 0 } = 0.3$ and by the recurrence relation, for all natural integer $n$ :
$$u _ { n + 1 } = 2 u _ { n } \left( 1 - u _ { n } \right)$$
This recurrence relation is written $u _ { n + 1 } = f \left( u _ { n } \right)$, where $f$ is the function defined on $\mathbb { R }$ by :
$$f ( x ) = 2 x ( 1 - x )$$

1. Prove that the function $f$ is strictly increasing on the interval $\left[ 0 ; \frac { 1 } { 2 } \right]$.
2. We admit that for all natural integer $n , 0 \leqslant u _ { n } \leqslant \frac { 1 } { 2 }$. Calculate $u _ { 1 }$ then perform a proof by induction to demonstrate that for all natural integer $n , u _ { n } \leqslant u _ { n + 1 }$.
3. Deduce that the sequence $( u _ { n } )$ is convergent.
4. Justify that the limit of the sequence $( u _ { n } )$ is equal to $\frac { 1 } { 2 }$.

The purpose of this part is to study a model of population evolution. In 2022, this population has 3000 individuals. We denote $P _ { n }$ the population size in thousands in the year $2022 + n$. Thus $P _ { 0 } = 3$. According to a model inspired by the Verhulst model, a Belgian mathematician of the XIX${}^{\mathrm{th}}$ century, we consider that, for all natural integer $n$ :
$$P _ { n + 1 } - P _ { n } = P _ { n } \left( 1 - b \times P _ { n } \right) , \text { where } b \text { is a strictly positive real number. }$$
The real number $b$ is a damping factor that allows us to account for the limited nature of the resources in the environment in which these individuals evolve.

1. In this question $b = 0$. a. Justify that the sequence $\left( P _ { n } \right)$ is a geometric sequence and specify its common ratio. b. Determine the limit of $P _ { n }$.
2. In this question $b = 0.2$. a. For all natural integer $n$, we set $v _ { n } = 0.1 \times P _ { n }$. Calculate $v _ { 0 }$ then show that, for all natural integer $n , v _ { n + 1 } = 2 v _ { n } \left( 1 - v _ { n } \right)$. b. In this model, justify that the population will stabilize around a value that you will specify.

Exercise 4 — 6 points Theme: sequences, functions Let $(u_n)$ be the sequence defined by $u_0 = -1$ and, for every natural number $n$: $$u_{n+1} = 0.9u_n - 0.3.$$

1. a. Prove by induction that, for all $n \in \mathbb{N}$, $u_n = 2 \times 0.9^n - 3$. b. Deduce that for all $n \in \mathbb{N}$, $-3 < u_n \leq -1$. c. Prove that the sequence $(u_n)$ is strictly decreasing. d. Prove that the sequence $(u_n)$ converges and specify its limit.
2. We propose to study the function $g$ defined on $]-3; -1]$ by: $$g(x) = \ln(0.5x + 1.5) - x.$$ a. Justify all the information given by the variations table of function $g$ (limits, variations, image of $-1$). b. Deduce that the equation $g(x) = 0$ has exactly one solution which we will denote $\alpha$ and for which we will give an interval of amplitude $10^{-3}$.
3. In the rest of the exercise, we consider the sequence $(v_n)$ defined for all $n \in \mathbb{N}$ by: $$v_n = \ln(0.5u_n + 1.5).$$ a. Using the formula given in question 1.a., prove that the sequence $v$ is arithmetic with common difference $\ln(0.9)$. b. Let $n$ be a natural number. Prove that $u_n = v_n$ if and only if $g(u_n) = 0$. c. Prove that there is no rank $k \in \mathbb{N}$ for which $u_k = \alpha$. d. Deduce that there is no rank $k \in \mathbb{N}$ for which $v_k = u_k$.

Exercise 4 — 5 points Theme: sequences, logarithm function, algorithms For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points:

1. Statement: The sequence $u$ defined for every natural integer $n$ by $u_{n} = \frac{(-1)^{n}}{n+1}$ is bounded.
2. Statement: Every bounded sequence is convergent.
3. Statement: Every increasing sequence tends to $+\infty$.
4. Let the function $f$ defined on $\mathbb{R}$ by $f(x) = \ln(x^{2} + 2x + 2)$. Statement: The function $f$ is convex on the interval $[-3; 1]$.
5. We consider the function mystery defined below which takes a list L of numbers as a parameter. We recall that len(L) returns the length, that is, the number of elements in the list $L$. \begin{verbatim} def mystery(L) : M = L[0] # We initialize M with the first element of the list L for i in range(len(L)) : if L[i] > M : M = L[i] return M \end{verbatim} Statement: The execution of mystery$([2, 3, 7, 0, 6, 3, 2, 0, 5])$ returns 7.

Consider the function $f$ defined on the interval $[ 0 ; 1 ]$ by $$f ( x ) = 2 x \mathrm { e } ^ { - x } .$$ It is admitted that the function $f$ is differentiable on the interval $[ 0 ; 1 ]$.

1. 1. [a.] Solve on the interval $[ 0 ; 1 ]$ the equation $f ( x ) = x$.
   2. [b.] Prove that, for all $x$ belonging to the interval $[ 0 ; 1 ]$, $$f ^ { \prime } ( x ) = 2 ( 1 - x ) \mathrm { e } ^ { - x } .$$
   3. [c.] Give the table of variations of the function $f$ on the interval $[ 0 ; 1 ]$.
   
   Consider the sequence $(u_n)$ defined by $u_0 = 0,1$ and for all natural integer $n$, $$u_{n+1} = f(u_n).$$
   
2. 1. [a.] Prove by induction that, for all natural integer $n$, $$0 \leqslant u_n < u_{n+1} \leqslant 1.$$
   2. [b.] Deduce that the sequence $(u_n)$ is convergent.
3. Prove that the limit of the sequence $(u_n)$ is $\ln(2)$.
4. 1. [a.] Justify that for all natural integer $n$, $\ln(2) - u_n$ is positive.
   2. [b.] It is desired to write a Python script that returns an approximate value of $\ln(2)$ by default to within $10^{-4}$, as well as the number of steps to achieve this. Copy and complete the script below so that it answers the problem posed. \begin{verbatim} def seuil() : n = 0 u=0.1 while ln (2) - u ...0.0001 : n=n+1 u=... return (u,n) \end{verbatim}
   3. [c.] Give the value of the variable $n$ returned by the function seuil().

Alain owns a swimming pool that contains $50\mathrm{~m}^3$ of water. Recall that $1\mathrm{~m}^3 = 1000\mathrm{~L}$. To disinfect the water, he must add chlorine. The chlorine level in the water, expressed in $\mathrm{mg}\cdot\mathrm{L}^{-1}$, is defined as the mass of chlorine per unit volume of water. Pool specialists recommend a chlorine level between 1 and $3\mathrm{~mg}\cdot\mathrm{L}^{-1}$. Under the action of the ambient environment, particularly ultraviolet rays, chlorine decomposes and gradually disappears. On Wednesday, June 19, he measures a chlorine level of $0.70\mathrm{~mg}\cdot\mathrm{L}^{-1}$.
Part A: study of a discrete model.
To maintain the chlorine level in his pool, Alain decides, starting from Thursday, June 20, to add 15 g of chlorine each day. It is admitted that this chlorine mixes uniformly in the pool water.

1. Justify that this addition of chlorine increases the level by $0.3\mathrm{~mg}\cdot\mathrm{L}^{-1}$.
2. For any natural number $n$, we denote by $v_n$ the chlorine level, in $\mathrm{mg}\cdot\mathrm{L}^{-1}$, obtained with this new protocol $n$ days after Wednesday, June 19. Thus $v_0 = 0.7$. It is admitted that for any natural number $n$, $$v_{n+1} = 0.92 v_n + 0.3.$$ a. Show by induction that for any natural number $n$, $\quad v_n \leqslant v_{n+1} \leqslant 4$. b. Show that the sequence $(v_n)$ is convergent and calculate its limit.
3. In the long term, will the chlorine level comply with the pool specialists' recommendation? Justify your answer.
4. Reproduce and complete the algorithm below written in Python language so that the function \texttt{alerte\_chlore} returns, when it exists, the smallest integer $n$ such that $v_n > s$. \begin{verbatim} def alerte_chlore(s) : n = 0 u=0.7 while...: n = ... u=... return n \end{verbatim}
5. What value is obtained by entering the instruction \texttt{alerte\_chlore(3)}? Interpret this result in the context of the exercise.

Part B: study of a continuous model.
Alain decides to call on a specialized engineering firm. This firm uses a continuous model to describe the chlorine level in the pool. In this model, for a duration $x$ (in days elapsed from Wednesday, June 19), $f(x)$ represents the chlorine level, in $\mathrm{mg}\cdot\mathrm{L}^{-1}$, in the pool. It is admitted that the function $f$ is a solution of the differential equation $(E): y' = -0.08y + \frac{q}{50}$, where $q$ is the quantity of chlorine, in grams, added to the pool each day.

1. Justify that the function $f$ is of the form $f(x) = C\mathrm{e}^{-0.08x} + \frac{q}{4}$ where $C$ is a real constant.
2. a. Express as a function of $q$ the limit of $f$ at $+\infty$. b. Recall that the chlorine level observed on Wednesday, June 19 is equal to $0.7\mathrm{~mg}\cdot\mathrm{L}^{-1}$. It is desired that the chlorine level stabilizes in the long term around $2\mathrm{~mg}\cdot\mathrm{L}^{-1}$. Determine the values of $C$ and $q$ so that these two conditions are satisfied.

We consider the function $g$ defined on the interval $[0; 1]$ by $$g(x) = 2x - x^2$$

1. Show that the function $g$ is strictly increasing on the interval $[0; 1]$ and specify the values of $g(0)$ and $g(1)$.

We consider the sequence $\left(u_n\right)$ defined by $\left\{\begin{array}{l} u_0 = \dfrac{1}{2} \\ u_{n+1} = g\left(u_n\right) \end{array}\right.$ for every natural number $n$.

1. Calculate $u_1$ and $u_2$.
2. Prove by induction that, for every natural number $n$, we have: $0 < u_n < u_{n+1} < 1$.
3. Deduce that the sequence $(u_n)$ is convergent.
4. Determine the limit $\ell$ of the sequence $(u_n)$.

We consider the sequence $(\nu_n)$ defined for every natural number $n$ by $\nu_n = \ln\left(1 - u_n\right)$.

1. Prove that the sequence $\left(v_n\right)$ is a geometric sequence with common ratio 2 and specify its first term.
2. Deduce an expression for $v_n$ as a function of $n$.
3. Deduce an expression for $u_n$ as a function of $n$ and find again the limit determined in question 5.
4. Copy and complete the Python script below so that it returns the rank $n$ from which the sequence exceeds 0.95. \begin{verbatim} def seuil() : n=0 u=0.5 while u < 0.95: n=... u=... return n \end{verbatim}

Exercise 3

Answer TRUE or FALSE to each of the following statements and justify your answer. Any answer without justification will not be taken into account in the grading. All questions in this exercise are independent.

1. Consider the sequence $( u _ { n } )$ defined for every non-zero natural number $n$ by $$u _ { n } = \frac { 25 + ( - 1 ) ^ { n } } { n }$$ Statement 1: The sequence $\left( u _ { n } \right)$ is divergent.
   
2. Consider the sequence $\left( w _ { n } \right)$ defined for every natural number $n$ by $\left\{ \begin{aligned} w _ { 0 } & = 1 \\ w _ { n + 1 } & = \frac { w _ { n } } { 1 + w _ { n } } \end{aligned} \right.$
   It is admitted that for every natural number $n , w _ { n } > 0$. Consider the sequence $( t _ { n } )$ defined for every natural number $n$ by $t _ { n } = \frac { k } { w _ { n } }$ where $k$ is a strictly positive real number. Statement 2: The sequence $\left( t _ { n } \right)$ is a strictly increasing arithmetic sequence.
   
3. Consider the sequence $\left( v _ { n } \right)$ defined for every natural number $n$ by $\left\{ \begin{array} { l l l } v _ { 0 } & = 1 \\ v _ { n + 1 } & = & \ln \left( 1 + v _ { n } \right) \end{array} \right.$ It is admitted that for every natural number $n , v _ { n } > 0$. Statement 3: The sequence $( v _ { n } )$ is decreasing.
   
4. Consider the sequence $\left( I _ { n } \right)$ defined for every natural number $n$ by $I _ { n } = \int _ { 1 } ^ { \mathrm { e } } [ \ln ( x ) ] ^ { n } \mathrm {~d} x$.
   Statement 4: For every natural number $n , I _ { n + 1 } = \mathrm { e } - ( n + 1 ) I _ { n }$.

For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points.

1. Let $( u _ { n } )$ be a sequence defined for all natural integer $n$ and satisfying the following relation: $$\text{for all natural integer } n , \frac { 1 } { 2 } < u _ { n } \leqslant \frac { 3 n ^ { 2 } + 4 n + 7 } { 6 n ^ { 2 } + 1 } .$$ Statement 1: $\lim _ { n \rightarrow + \infty } u _ { n } = \frac { 1 } { 2 }$.
   
2. Let $h$ be a function defined and differentiable on the interval $[-4;4]$. The graphical representation $\mathscr { C } _ { h ^ { \prime } }$ of its derivative function $h ^ { \prime }$ is given below. Statement 2: The function $h$ is convex on $[ - 1 ; 3]$.
   
3. The code of a building is composed of 4 digits (which may be identical) followed by two distinct letters among A, B and C (example: 1232BA). Statement 3: There exist 20634 codes that contain at least one 0.
   
4. We consider the function $f$ defined on $] 0 ; + \infty [$ by $f ( x ) = x \ln x$. Statement 4: The function $f$ is a solution on $] 0 ; + \infty [$ of the differential equation $$x y ^ { \prime } - y = x .$$

Exercise 3 (5 points)

For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points. The five questions in this exercise are independent.

1. Consider a sequence ( $t _ { n }$ ) satisfying the recurrence relation: $$\text { for all natural integer } n , t _ { n + 1 } = - 0.8 t _ { n } + 18 .$$ Statement 1: The sequence ( $w _ { n }$ ) defined for all natural integer $n$ by $w _ { n } = t _ { n } - 10$ is geometric.
   
2. Consider a sequence ( $S _ { n }$ ) that satisfies for all non-zero natural integer $n$: $$3 n - 4 \leqslant S _ { n } \leqslant 3 n + 4 .$$ The sequence ( $u _ { n }$ ) is defined, for all non-zero natural integer $n$, by: $u _ { n } = \frac { S _ { n } } { n }$. Statement 2: The sequence ( $u _ { n }$ ) converges.
   
3. Consider the sequence $\left( v _ { n } \right)$ defined by: $$v _ { 1 } = 2 \text { and for all natural integer } n \geqslant 1 , v _ { n + 1 } = 2 - \frac { 1 } { v _ { n } } .$$ Statement 3: For all natural integer $n \geqslant 1 , v _ { n } = \frac { n + 1 } { n }$.
   
4. Consider the sequence ( $u _ { n }$ ) defined for all natural integer $n$ by $u _ { n } = \mathrm { e } ^ { n } - n$. Statement 4: The sequence $\left( u _ { n } \right)$ converges.
   
5. Consider the sequence ( $u _ { n }$ ) defined using the script written below in Python language, which returns the value of $u _ { n }$. \begin{verbatim} def u(n) : valeur = 2 for k in range(n) : valeur = 0.5 * (valeur + 2/valeur) return valeur \end{verbatim} We admit that ( $u _ { n }$ ) is decreasing and satisfies for all natural integer $n$: $$\sqrt { 2 } \leqslant u _ { n } \leqslant 2 .$$ Statement 5: The sequence $\left( u _ { n } \right)$ converges to $\sqrt { 2 }$.

Consider the sequence $(u_n)$ defined by: $$u_0 = 8 \text{ and for every natural number } n,\quad u_{n+1} = u_n - \ln\left(\frac{u_n}{4}\right).$$

1. a. Give the values rounded to the nearest hundredth of $u_1$ and $u_2$. b. Consider the mystery function defined below in Python. We admit that, for every strictly positive real number $a$, $\log(a)$ returns the value of the natural logarithm of $a$. \begin{verbatim} def mystery(k) : u = 8 S = 0 for i in range(k) : S = S + u u = u - log( u / 4) return S \end{verbatim} The execution of \texttt{mystery(10)} returns 58.44045206721732. What does this result represent? c. Modify the previous function so that it returns the average of the first $k$ terms of the sequence $(u_n)$.
   
2. Consider the function $f$ defined and differentiable on $]0; +\infty[$ by: $$f(x) = x - \ln\left(\frac{x}{4}\right).$$ Study the variations of $f$ on $]0; +\infty[$ and draw its variation table. The exact value of the minimum of $f$ on $]0; +\infty[$ will be specified. Limits are not required.
   In the rest of the exercise, it will be noted that for every natural number $n$, $u_{n+1} = f(u_n)$.
   
3. a. Prove, by induction, that for every natural number $n$, we have: $$1 \leqslant u_{n+1} \leqslant u_n.$$ b. Deduce that the sequence $(u_n)$ converges to a real limit.
   Let $\ell$ denote the value of this limit. c. Solve the equation $f(x) = x$. d. Deduce the value of $\ell$.

For each of the following statements, specify whether it is true or false then justify the answer given. Any answer without justification will not be taken into account.

1. Statement 1: Any decreasing sequence and bounded below by 0 converges to 0.
2. We consider a sequence $(u_n)$ defined on $\mathbb{N}$ such that, for every integer $n$, we have $$u_n \leq \frac{-9^n + 3^n}{7^n}.$$ Statement 2: $\lim_{n \rightarrow +\infty} u_n = -\infty$.
3. We consider the following function written in Python language: \begin{verbatim} def terme(N) : U = 1 for i in range(N) : U = U + i return U \end{verbatim} Statement 3: terme(4) returns the value 7.
4. During a competition, the winner has a choice between two prizes:
   • Prize A: they receive 1000 euros per day for 15 days;
   • Prize B: they receive 1 euro on day 1, 2 euros on day 2, 4 euros on day 3 and for 15 days the sum received doubles each day.
   Statement 4: The value of prize A is higher than the value of prize B.
5. We consider the sequence $(v_n)$ defined for every integer $n \geq 1$ by $$v_n = \int_1^n \ln x \mathrm{~d}x.$$ Statement 5: The sequence $(v_n)$ is increasing.

Part A
We consider the function $f$ defined on the interval $[0; +\infty[$ by $$f(x) = \sqrt{x+1}.$$ We admit that this function is differentiable on this same interval.

1. Prove that the function $f$ is increasing on the interval $[0; +\infty[$.
2. Prove that for every real number $x$ belonging to the interval $[0; +\infty[$: $$f(x) - x = \frac{-x^2 + x + 1}{\sqrt{x+1} + x}.$$
3. Deduce from this that on the interval $[0; +\infty[$ the equation $f(x) = x$ admits as unique solution: $$\ell = \frac{1+\sqrt{5}}{2}.$$

Part B
We consider the sequence $(u_n)$ defined by $u_0 = 5$ and for every natural number $n$, by $u_{n+1} = f(u_n)$ where $f$ is the function studied in part A. We admit that the sequence with general term $u_n$ is well defined for every natural number $n$.

1. Prove by induction that for every natural number $n$, we have $$1 \leqslant u_{n+1} \leqslant u_n.$$
2. Deduce from this that the sequence $(u_n)$ converges.
3. Prove that the sequence $(u_n)$ converges to $\ell = \frac{1+\sqrt{5}}{2}$.
4. We consider the Python script below: \begin{verbatim} from math import * def seuil(n): u=5 i=0 while abs(u-l)>=10**(-n): u=sqrt(u+1) i=i+1 return(i) \end{verbatim} We recall that the command $\mathbf{abs}(\mathbf{x})$ returns the absolute value of $x$.
   1. [a.] Give the value returned by \texttt{seuil(2)}.
   2. [b.] The value returned by \texttt{seuil(4)} is 9. Interpret this value in the context of the exercise.

The objective of this exercise is to conjecture in Part A and then prove in Part B the behavior of a sequence. The two parts can, however, be treated independently. We consider the sequence ( $u _ { n }$ ) defined by $u _ { 0 } = 3$ and for all $n \in \mathbb { N }$ :
$$u _ { n + 1 } = \frac { 4 } { 5 - u _ { n } } .$$
Part A

1. Copy and complete the following Python function suite(n) which takes as parameter the rank $n$ and returns the value of the term $u _ { n }$.

\begin{verbatim} def suite(n): u = ... for i in range(n) : ... return u \end{verbatim}

1. The execution of suite(2) returns 1.3333333333333333 .

Perform a calculation to verify and explain this output.
3. Using the outputs below, make a conjecture about the direction of variation and a conjecture about the convergence of the sequence ( $u _ { n }$ ).
\begin{verbatim} > suite(2) 1.3333333333333333 >> suite(5) 1.0058479532163742 >> suite(10) 1.0000057220349845 > suite(20) 1.000000000005457 \end{verbatim}
Part B We consider the function $f$ defined and differentiable on the interval $] - \infty ; 5 [$ by:
$$f ( x ) = \frac { 4 } { 5 - x }$$
Thus, the sequence ( $u _ { n }$ ) is defined by $u _ { 0 } = 3$ and for all $n \in \mathbb { N } , u _ { n + 1 } = f \left( u _ { n } \right)$.

1. Show that the function $f$ is increasing on the interval $] - \infty$; 5[.
2. Prove by induction that for every natural integer $n$ we have:

$$1 \leqslant u _ { n + 1 } \leqslant u _ { n } \leqslant 4 .$$

1. a. Let $x$ be a real number in the interval $] - \infty$; 5[. Prove the following equivalence:
   $$f ( x ) = x \Longleftrightarrow x ^ { 2 } - 5 x + 4 = 0 .$$
   b. Solve $f ( x ) = x$ in the interval $] - \infty$; 5[.
2. Prove that the sequence ( $u _ { n }$ ) is convergent. Determine its limit.
3. Would the behavior of the sequence be identical by choosing as initial term $u _ { 0 } = 4$ instead of $u _ { 0 } = 3$ ?

Throughout the exercise, probabilities will be rounded to $10^{-3}$ if necessary.
A binary datum is data that can only take two values: 0 or 1. Data of this type is transmitted successively from one machine to another. Each machine transmits the received data either faithfully, that is, by transmitting the information as it received it (1 becomes 1 and 0 becomes 0), or in the opposite way (1 becomes 0 and 0 becomes 1). The transmission is faithful in $90\%$ of cases, and therefore opposite in $10\%$ of cases. Throughout the exercise, the first machine always receives the value 1.
For any natural integer $n \geqslant 1$, we denote:

• $V_n$ the event: ``the $n$-th machine holds the value 1'';
• $\overline{V_n}$ the event: ``the $n$-th machine holds the value 0''.

Part A

1. a. Copy and complete the probability tree. b. Prove that $P(V_3) = 0{,}82$ and interpret this result in the context of the exercise. c. Given that the third machine received the value 1, calculate the probability that the second machine also received the value 1.
2. For any natural integer $n \geqslant 1$, we denote $p_n = P(V_n)$. The first machine received the value 1, so $p_1 = 1$. a. Prove that for any natural integer $n \geqslant 1$: $$p_{n+1} = 0{,}8\, p_n + 0{,}1.$$ b. Prove by induction that for any natural integer $n \geqslant 1$, $$p_n = 0{,}5 \times 0{,}8^{n-1} + 0{,}5.$$ c. Calculate the limit of $p_n$ as $n$ tends to infinity. Interpret this result in the context of the exercise.

Part B
To model in Python language the transmission of the binary datum described at the beginning of the exercise, we consider the simulation function which takes as a parameter a natural integer $n$ which represents the number of transmissions carried out from one machine to another, and which returns the list of successive values of the binary datum. The incomplete script of this function is given below. We recall that the instruction rand() returns a random number from the interval $[0; 1[$.
\begin{verbatim} def simulation(n): donnee = 1 liste = [donnee] for k in range(n): if rand() <0.1 donnee = 1 - donnee liste.append(donnee) return liste \end{verbatim}
For example, simulation(3) can return $[1, 0, 0, 1]$.

1. Determine the role of the instructions on lines 5 and 6 of the algorithm above.
2. Calculate the probability that simulation(4) returns the list $[1,1,1,1,1]$ and the probability that simulation(6) returns the list $[1,0,1,0,0,1,1]$.