Consider the sequence $(u_n)$ defined for every natural number $n$ by: $$u_n = \mathrm{e}^{2n+1}$$ The sequence $(u_n)$ is: a. arithmetic with common difference 2; b. geometric with common ratio e; c. geometric with common ratio $\mathrm{e}^2$; d. convergent to e.

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.

Let the sequence $(u_n)$ defined by $u_0 = 0$ and, for all $n \in \mathbb{N}$, $$u_{n+1} = 5u_n - 8n + 6.$$

1. Calculate $u_1$ and $u_2$.
2. Let $n$ be a natural number. Copy and complete the function \texttt{suite\_u} with argument \texttt{n} below, written in Python language, so that it returns the value of $u_n$. \begin{verbatim} def suite_u(n) : u = ... for i in range(1,n+1) : u = ... return u \end{verbatim}
3. a. Prove by induction that, for all $n \in \mathbb{N}$, $u_n \geqslant 2n$. b. Deduce the limit of the sequence $(u_n)$. c. Let $p \in \mathbb{N}^*$. Why can we assert that there exists at least one integer $n_0$ such that, for all natural integers $n$ satisfying $n \geqslant n_0$, $u_n \geqslant 10^p$?
4. Prove that the sequence $(u_n)$ is increasing.
5. We consider the sequence $(v_n)$, defined for all $n \in \mathbb{N}$, by $v_n = u_n - 2n + 1$. a. Below the function \texttt{suite\_u} above, we have written the function \texttt{suite\_v} below: \begin{verbatim} def suite_v(n): L = [] for i in range(n+1) : L.append(suite_u(i) - 2*i + 1) return L \end{verbatim} The command ``L.append'' allows us to add, in the last position, an element to the list $L$. When we enter \texttt{suite\_v(5)} in the console, we obtain the following display: $$\begin{aligned} & \ggg \text{suite\_v}(5) \\ & [1, 5, 25, 125, 625, 3125] \end{aligned}$$ Conjecture, for all natural integer $n$, the expression of $v_{n+1}$ as a function of $v_n$. Prove this conjecture. b. Deduce, for all natural integer $n$, the explicit form of $u_n$ as a function of $n$.

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. $n \leqslant 10000$.

We consider the sequence $(u_n)$ defined by: $$\left\{\begin{aligned} u_1 &= \frac{1}{\mathrm{e}} \\ u_{n+1} &= \frac{1}{\mathrm{e}}\left(1 + \frac{1}{n}\right)u_n \text{ for all integer } n \geqslant 1. \end{aligned}\right.$$

1. Calculate the exact values of $u_2$ and $u_3$. Details of calculations should be shown.
2. We consider a function written in Python language which, for a given natural integer $n$, displays the term $u_n$. Complete lines $L_2$ and $L_4$ of this program. \begin{verbatim} L1 def u(n): L2 ................. L3 for i in range(1, n): L4 u=................. L5 return u \end{verbatim}
3. We admit that all terms of the sequence $(u_n)$ are strictly positive. a. Show that for all non-zero natural integer $n$, we have: $1 + \dfrac{1}{n} \leqslant \mathrm{e}$. b. Deduce that the sequence $(u_n)$ is decreasing. c. Is the sequence $(u_n)$ convergent? Justify your answer.
4. a. Show by induction that for all non-zero natural integer, we have: $u_n = \dfrac{n}{\mathrm{e}^n}$. b. Deduce, if it exists, the limit of the sequence $(u_n)$.

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.

For questions 3. and 4., consider the sequence $(u_n)$ defined on $\mathbb{N}$ by: $$u_0 = 15 \text{ and for every natural number } n : u_{n+1} = 1{,}2\, u_n + 12.$$
Consider the sequence $(v_n)$ defined on $\mathbb{N}$ by: $v_n = u_n + 60$.
The sequence $(v_n)$ is: a. a decreasing sequence; b. a geometric sequence with common ratio 1,2; c. an arithmetic sequence with common difference 60; d. a sequence that is neither geometric nor arithmetic.

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.

This exercise contains 5 statements. For each statement, answer TRUE or FALSE by justifying the answer. Any lack of justification or incorrect justification will not be taken into account in the grading.
Part 1
We consider the sequence $(u_n)$ defined by: $$u_0 = 10 \text{ and for all natural integer } n,\ u_{n+1} = \frac{1}{3}u_n + 2.$$

1. Statement 1: The sequence $(u_n)$ is decreasing and bounded below by 0.
2. Statement 2: $\lim_{n \rightarrow +\infty} u_n = 0$.
3. Statement 3: The sequence $(v_n)$ defined for all natural integer $n$ by $v_n = u_n - 3$ is geometric.

Part 2
We consider the differential equation $(E): y' = \frac{3}{2}y + 2$ with unknown $y$, a function defined and differentiable on $\mathbb{R}$.

1. Statement 4: There exists a constant function that is a solution to the differential equation $(E)$.
2. In an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath})$ we denote $\mathscr{C}_f$ the representative curve of the function $f$ solution of $(E)$ such that $f(0) = 0$. Statement 5: The tangent line at the point with abscissa 1 of $\mathscr{C}_f$ has slope $2\mathrm{e}^{\frac{3}{2}}$.

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.

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 $u _ { 0 } = 3$ ?

We consider the numerical sequence $(u_n)$ defined by its first term $u_0 = 2$ and for every natural number $n$, by: $$u_{n+1} = \frac{2u_n + 1}{u_n + 2}$$ We admit that the sequence $(u_n)$ is well defined.

1. Calculate the term $u_1$.
2. We define the sequence $(a_n)$ for every natural number $n$, by: $$a_n = \frac{u_n}{u_n - 1}$$ We admit that the sequence $(a_n)$ is well defined. a. Calculate $a_0$ and $a_1$. b. Prove that, for every natural number $n$, $a_{n+1} = 3a_n - 1$. c. Prove by induction that, for every natural number $n$ greater than or equal to 1, $$a_n \geqslant 3n - 1$$ d. Deduce the limit of the sequence $(a_n)$.
3. We wish to study the limit of the sequence $(u_n)$. a. Prove that for every natural number $n$, $u_n = \frac{a_n}{a_n - 1}$. b. Deduce the limit of the sequence $(u_n)$.
4. We admit that the sequence $(u_n)$ is decreasing.
   We consider the following program written in Python: \begin{verbatim} def algo(p): u=2 n=0 while u-1>p: u=(2*u+1)/(u+2) n=n+1 return (n,u) \end{verbatim} a. Interpret the values $n$ and u returned by the call to the function algo(p) in the context of the exercise. b. Give, without justification, the value of $n$ for $p = 0.001$.

One of the objectives of this exercise is to determine an approximation of the real number $\ln ( 2 )$, using one of the methods of the English mathematician Henry Briggs in the XVI${}^{\text{th}}$ century.
We denote by ( $u _ { n }$ ) the sequence defined by:
$$u _ { 0 } = 2 \quad \text { and, for every natural number } n , \quad u _ { n + 1 } = \sqrt { u _ { n } }$$

Part A

1. a. Give the exact value of $u _ { 1 }$ and $u _ { 2 }$. b. Make a conjecture, using a calculator, about the direction of variation and the possible limit of the sequence.
2. a. Show by induction that for every natural number $n , \quad 1 \leqslant u _ { n + 1 } \leqslant u _ { n }$. b. Deduce that the sequence ( $u _ { n }$ ) is convergent. c. Solve in the interval [ $0 ; + \infty$ [ the equation $\sqrt { x } = x$. d. Determine, by justifying, the limit of the sequence $\left( u _ { n } \right)$.

Part B

We denote by ( $\nu _ { n }$ ) the sequence defined for every natural number $n$ by $\nu _ { n } = \ln \left( u _ { n } \right)$.

1. a. Prove that the sequence ( $v _ { n }$ ) is geometric with common ratio $\frac { 1 } { 2 }$. b. Express $v _ { n }$ as a function of $n$, for every natural number $n$. c. Deduce that, for every natural number $n , \quad \ln ( 2 ) = 2 ^ { n } \ln \left( u _ { n } \right)$.
2. We have traced below in an orthonormal coordinate system the curve $\mathscr { C }$ of the function ln and the tangent T to the curve $\mathscr { C }$ at the point with abscissa 1. An equation of the line T is $y = x - 1$. The points $\mathrm { A } _ { 0 } , \mathrm {~A} _ { 1 } , \mathrm {~A} _ { 2 }$ have abscissas $u _ { 0 } , u _ { 1 }$ and $u _ { 2 }$ respectively and ordinate 0. We decide to take $x - 1$ as an approximation of $\ln ( x )$ when $x$ belongs to the interval $] 0,99 ; 1,01 [$. a. Using a calculator, determine the smallest natural number $k$ such that $u _ { k }$ belongs to the interval $] 0,99 ; 1,01 [$ and give an approximate value of $u _ { k }$ to $10 ^ { - 5 }$ near. b. Deduce an approximation of $\ln \left( u _ { k } \right)$. c. Deduce from questions 1.c. and 2.b. of Part B an approximation of $\ln ( 2 )$.
3. We generalize the previous method to any real number $a$ strictly greater than 1.
   Copy and complete the algorithm below so that the call Briggs(a) returns an approximation of $\ln ( a )$.
   We recall that the instruction in Python language sqrt (a) corresponds to $\sqrt { a }$.
   \begin{verbatim} from math import* def Briggs(a): n = 0 while a >= 1.01: a = sqrt(a) n = n+1 L =... return L \end{verbatim}

Consider the sequences $\left(v_n\right)$ and $\left(w_n\right)$ defined for every natural integer $n$ by:
$$\left\{ \begin{array}{ll} v_0 &= \ln(4) \\ v_{n+1} &= \ln\left(-1 + 2\mathrm{e}^{v_n}\right) \end{array} \quad \text{and} \quad w_n = -1 + \mathrm{e}^{v_n} \right.$$
We admit that the sequence $\left(v_n\right)$ is well defined and strictly positive.

1. Give the exact values of $v_1$ and $w_0$.
2. a. Among the three formulas below, choose the formula which, entered in cell B3 then copied downward, will allow you to obtain the values of the sequence $(v_n)$ in column B.
   

   Formula 1	$\mathrm{LN}\left(-1 + 2^*\operatorname{EXP}(\mathrm{B}2)\right)$
   Formula 2	$=\mathrm{LN}\left(-1 + 2^*\operatorname{EXP}(\mathrm{B}2)\right)$
   Formula 3	$=\mathrm{LN}\left(-1 + 2^*\operatorname{EXP}(\mathrm{A}2)\right)$

   
   b. Conjecture the direction of variation of the sequence $\left(v_n\right)$. c. Using a proof by induction, validate your conjecture concerning the direction of variation of the sequence $(v_n)$.
3. a. Prove that the sequence $(w_n)$ is geometric. b. Deduce that for every natural integer $n$, $v_n = \ln\left(1 + 3 \times 2^n\right)$. c. Determine the limit of the sequence $\left(v_n\right)$.
4. Justify that the following algorithm written in Python language returns a result regardless of the choice of the value of the number S. \begin{verbatim} from math import* def seuil(S): V=ln(4) n=0 while V < S : n=n+1 V=ln(2*exp(V)-1) return(n) \end{verbatim}

Exercise 3
The purpose of this exercise is to study the convergence of two sequences towards the same limit.
Part A
Consider the function $f$ defined on $[2;+\infty[$ by $$f(x) = \sqrt{3x-2}$$

1. Justify the elements of the variation table below:
   

   $x$	2	$+\infty$
   		$+\infty$
   $f(x)$
   	2

   
   We admit that the sequence $(u_n)$ satisfying $u_0 = 6$ and, for all natural number $n$, $u_{n+1} = f(u_n)$ is well defined.
   
2. 1. [a.] Prove by induction that, for all natural number $n$: $2 \leqslant u_{n+1} \leqslant u_n \leqslant 6$.
   2. [b.] Deduce that the sequence $(u_n)$ converges.
3. We call $\ell$ the limit of $(u_n)$.
   We admit that it is a solution of the equation $f(x) = x$. Determine the value of $\ell$.
   
4. Consider the rank function written below in Python language.
   We recall that $\operatorname{sqrt}(x)$ returns the square root of the number $x$.
   \begin{verbatim} from math import * def rang(a) : u=6 n=0 while u >= a : u = sqrt(3*u - 2) n = n+1 return n \end{verbatim}
   
   1. [a.] Why can we affirm that rang(2.000001) returns a value?
   2. [b.] For which values of the parameter $a$ does the instruction rang($a$) return a result?

Part B
We admit that the sequence $(v_n)$ satisfying $v_0 = 6$ and, for all natural number $n$, $v_{n+1} = 3 - \dfrac{2}{v_n}$ is well defined.

1. Calculate $v_1$.
2. For all natural number $n$, we admit that $v_n \neq 2$ and we set: $$w_n = \frac{v_n - 1}{v_n - 2}$$
   1. [a.] Prove that the sequence $(w_n)$ is geometric with ratio 2 and specify its first term $w_0$.
   2. [b.] We admit that, for all natural number $n$, $$w_n - 1 = \frac{1}{v_n - 2}$$ Deduce that, for all natural number $n$, $$v_n = 2 + \frac{1}{1{,}25 \times 2^n - 1}$$
   3. [c.] Calculate the limit of $(v_n)$.
3. Determine the smallest natural number $n$ for which $v_n < 2{,}01$ by solving the inequality.

Part C
Using the previous parts, determine the smallest integer $N$ such that for all $n \geqslant N$, the terms $v_n$ and $u_n$ belong to the interval $]1{,}99;2{,}01[$.

We consider the sequence $(u_n)$ defined by $u_0 = 5$ and, for all natural integers $n$: $$u_{n+1} = 2 + \ln\left(u_n^2 - 3\right)$$ We admit that this sequence is well defined.
Part A: Exploitation of Python programs

1. Copy and complete the Python script below so that \texttt{suite(k)} which takes a natural integer $k$ as parameter returns the list of the first $k$ values of the sequence $(u_n)$.
   Remark: We specify that, for any strictly positive real number $a$, $\log(a)$ returns the value of the natural logarithm of $a$.
   \begin{verbatim} def suite(k): L = [] u = 5 for i in range(......): L.append(u) u=............ return(......) \end{verbatim}
   
2. We executed \texttt{suite(9)} below. Make two conjectures: one on the direction of variation of the sequence $(u_n)$ and another on its possible convergence.
   \begin{verbatim} >>> suite(9) [ 5, 5.091042453358316, 5.131953749864703, 5.150037910978289, 5.157974010229213, 5.1614456706362954, 5.162962248594583, 5.163624356938671, 5.163913344065642] \end{verbatim}
   
3. We then created the function \texttt{mystere(n)} given below and executed \texttt{mystere(10000)}, which returned 1. Does this output contradict the conjecture made about the direction of variation of the sequence $(u_n)$? Justify.
   \begin{verbatim} def mystere(n): L = suite(n) c = 1 for i in range(n - 1): if L[i] > L[i + 1]: c = 0 return c
   >>> mystere(10000) 1 \end{verbatim}

Part B: Study of the convergence of the sequence $(u_n)$
We consider the function $g$ defined on $[2; +\infty[$ by: $$g(x) = 2 + \ln\left(x^2 - 3\right)$$ We admit that $g$ is differentiable on $[2; +\infty[$ and we denote $g'$ its derivative function.

1. Prove that the function $g$ is increasing on $[2; +\infty[$.
2. a. Prove by induction that, for all natural integers $n$: $$4 \leqslant u_n \leqslant u_{n+1} \leqslant 6$$ b. Deduce that the sequence $(u_n)$ converges.

Part C: Study of the limit value
We consider the function $f$ defined on $[2; +\infty[$ by: $$f(x) = 2 + \ln\left(x^2 - 3\right) - x$$ We admit that $f$ is differentiable on $[2; +\infty[$ and we denote $f'$ its derivative function. We give the following variation table of $f$. No justification is requested.

$x$	2	3	$+\infty$
		$\ln(6) - 1$
$f(x)$
	0		$-\infty$

1. a. Show that the equation $f(x) = 0$ has exactly two solutions on $[2; +\infty[$ which we will denote $\alpha$ and $\beta$ with $\alpha < \beta$. b. Give the exact value of $\alpha$ and an approximate value to $10^{-3}$ of $\beta$.
2. Let $\ell$ be the limit of the sequence $(u_n)$. Justify that $f(\ell) = 0$ and determine $\ell$.