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

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

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

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

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

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

   & &

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

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

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

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

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

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

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

For any natural integer $n$, we denote:

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

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

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

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

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

\begin{tabular}{l} Variables:

Input: Processing:

Output:

&

$k$ and $p$ are natural integers

$u$ is a real number

Ask for the value of $p$

Assign to $u$ the value 5

For $k$ varying from 1 to $p$

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

End for

Display $u$

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

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

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

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

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

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

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

   \hline Initialization: &

   $D$ takes the value 300

   $A$ takes the value 450

   Enter the value of $n$

   \hline Processing: &

   For $k$ varying from 1 to $n$

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

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

   End for

   \hline Output: &

   Display $D$

   Display $A$

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

(For candidates who have followed the specialization course)
An organization offers online foreign language learning. Two levels are presented: beginner or advanced. At the beginning of each month, an internet user can register, unregister or change level. At the beginning of month 0, there were 300 internet users at the beginner level and 450 at the advanced level. Each month, half of the beginners move to the advanced level, the other half remain at the beginner level and half of the advanced users who have completed their training unregister from the site. Each month, 100 new internet users register as beginners and 70 as advanced. This situation is modeled by two sequences of real numbers $( d _ { n } )$ and $( a _ { n } )$. For every natural number $n$, $d _ { n }$ and $a _ { n }$ are respectively approximations of the number of beginners and the number of advanced users at the beginning of month $n$. For every natural number $n$, we denote by $U _ { n }$ the column matrix $\binom { d _ { n } } { a _ { n } }$. We set $d _ { 0 } = 300$, $a _ { 0 } = 450$ and, for every integer $n \geqslant 0$
$$\left\{ \begin{aligned} d _ { n + 1 } & = \frac { 1 } { 2 } d _ { n } + 100 \\ a _ { n + 1 } & = \frac { 1 } { 2 } d _ { n } + \frac { 1 } { 2 } a _ { n } + 70 \end{aligned} \right.$$

1. a. Justify the equality $a _ { n + 1 } = \frac { 1 } { 2 } d _ { n } + \frac { 1 } { 2 } a _ { n } + 70$ in the context of the exercise. b. Determine the matrices $A$ and $B$ such that for every natural number $n$, $$U _ { n + 1 } = A U _ { n } + B$$
2. Prove by induction that for every natural number $n \geqslant 1$, we have $$A ^ { n } = \left( \frac { 1 } { 2 } \right) ^ { n } \left( I _ { 2 } + n T \right) \quad \text { where } T = \left( \begin{array} { l l } 0 & 0 \\ 1 & 0 \end{array} \right) \quad \text { and } I _ { 2 } = \left( \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right) .$$
3. a. Determine the matrix $C$ that satisfies the equality $C = A C + B$. b. For every integer $n \geqslant 0$, we set $V _ { n } = U _ { n } - \binom { 200 } { 340 }$.

Exercise 2

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

1. Here is an extract from a spreadsheet:
   

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

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

A company produces bacteria for industry. In the laboratory, it was measured that, in an appropriate nutrient medium, the mass of these bacteria, measured in grams, increases by $20\%$ in one day. The company implements the following industrial process. In a vat of nutrient medium, 1 kg of bacteria is initially introduced. Then, each day, at a fixed time, the nutrient medium in the vat is replaced. During this operation, 100 g of bacteria are lost. The company's objective is to produce 30 kg of bacteria.
Part A: first model - with a sequence
The evolution of the bacterial population in the vat is modeled by the sequence $(u _ { n })$ defined as follows:
$$u _ { 0 } = 1000 \text{ and, for all natural integers } n , u _ { n + 1 } = 1.2 u _ { n } - 100 .$$

1. a. Explain how this model corresponds to the situation described in the problem. You will specify in particular what $u _ { n }$ represents. b. The company wants to know after how many days the mass of bacteria will exceed 30 kg. Using a calculator, give the answer to this problem. c. We can also use the following algorithm to answer the problem posed in the previous question. Copy and complete this algorithm.
   

   Variables	$u$ and $n$ are numbers
   Processing	\begin{tabular}{l} $u$ takes the value 1000
   $n$ takes the value 0
   While $\_\_\_\_$ do
   $u$ takes the value $\_\_\_\_$ $n$ takes the value $n + 1$
   End While

   \hline Output & Display .......... \hline \end{tabular}
   
2. a. Prove by induction that, for all natural integers $n$, $u _ { n } \geqslant 1000$. b. Prove that the sequence $( u _ { n } )$ is increasing.
3. We define the sequence $( v _ { n } )$ by: for all natural integers $n$, $v _ { n } = u _ { n } - 500$. a. Prove that the sequence $( v _ { n } )$ is a geometric sequence. b. Express $v _ { n }$, then $u _ { n }$, as a function of $n$. c. Determine the limit of the sequence $( u _ { n } )$.

Part B: second model - with a function
It is observed that in practice, the mass of bacteria in the vat will never exceed 50 kg. This leads to studying a second model in which the mass of bacteria is modeled by the function $f$ defined on $[ 0 ; +\infty[$ by:
$$f ( t ) = \frac { 50 } { 1 + 49 \mathrm { e } ^ { - 0.2 t } }$$
where $t$ represents time expressed in days and where $f ( t )$ represents the mass, expressed in kg, of bacteria at time $t$.

1. a. Calculate $f ( 0 )$. b. Prove that, for all real $t \geqslant 0$, $f ( t ) < 50$. c. Study the monotonicity of the function $f$. d. Determine the limit of the function $f$ as $t \to + \infty$.
2. Interpret the results of question 1 in the context of the problem.
3. Using this model, we seek to determine after how many days the mass of bacteria will exceed 30 kg. Solve the inequality with unknown $t$: $f ( t ) > 30$. Deduce the answer to the problem.

Part C: quality control
Bacteria can be of two types: type A, which effectively produces a protein useful to industry, and type B, which does not produce it and is therefore commercially useless. The company claims that $80\%$ of the bacteria produced are of type A. To verify this claim, a laboratory analyzes a random sample of 200 bacteria at the end of production. The analysis shows that 146 of them are of type A. Should the company's claim be questioned?

In a factory, a kiln bakes ceramics at a temperature of $1000^{\circ}\mathrm{C}$. At the end of baking, it is turned off and cools down. The temperature of the kiln is expressed in degrees Celsius (${}^{\circ}\mathrm{C}$). The kiln door can be opened safely for the ceramics as soon as its temperature is below $70^{\circ}\mathrm{C}$.
For a natural integer $n$, we denote $T_n$ the temperature in degrees Celsius of the kiln after $n$ hours have elapsed from the moment it was turned off. We therefore have $T_0 = 1000$. The temperature $T_n$ is calculated by the following algorithm:
\begin{verbatim} T←1000 For i going from 1 to n T←0.82 x T+3.6 End For \end{verbatim}

1. Determine the temperature of the kiln, rounded to the nearest unit, after 4 hours of cooling.
2. Prove that, for every natural integer $n$, we have: $T_n = 980 \times 0.82^n + 20$.
3. After how many hours can the kiln be opened safely for the ceramics?

In May 2020, a company chose to develop telework. It proposed to its 5000 employees in France to choose between telework and working at the company's premises. In May 2020, only 200 of them chose telework. Each month, since the implementation of this measure, $85\%$ of those who had chosen telework the previous month choose to continue, and each month, 450 additional employees choose telework. The number of company employees working from home is modeled by the sequence $(a_n)$. The term $a_n$ designates an estimate of the number of employees working from home in the $n$-th month after May 2020. Thus $a_0 = 200$.
Part A:

1. Calculate $a_1$.
2. Justify that for every natural number $n$, $a_{n+1} = 0.85a_n + 450$.
3. Consider the sequence $(v_n)$ defined for every natural number $n$ by: $v_n = a_n - 3000$. a. Prove that the sequence $(v_n)$ is a geometric sequence with common ratio 0.85. b. Express $v_n$ as a function of $n$ for every natural number $n$. c. Deduce that, for every natural number $n$, $a_n = -2800 \times 0.85^n + 3000$.
4. Determine the number of months after which the number of teleworkers will be strictly greater than 2500, after the implementation of this measure in the company.

Part B: The company's managers modeled the number of employees satisfied with this system using the sequence $(u_n)$ defined by $u_0 = 1$ and, for every natural number $n$, $$u_{n+1} = \frac{5u_n + 4}{u_n + 2}$$ where $u_n$ denotes the number of thousands of employees satisfied with this new measure after $n$ months following May 2020.

1. Prove that the function $f$ defined for all $x \in [0;+\infty[$ by $f(x) = \dfrac{5x+4}{x+2}$ is strictly increasing on $[0;+\infty[$.
2. a. Prove by induction that for every natural number $n$: $$0 \leqslant u_n \leqslant u_{n+1} \leqslant 4.$$ b. Justify that the sequence $(u_n)$ is convergent.
3. We admit that for every natural number $n$, $$0 \leqslant 4 - u_n \leqslant 3 \times \left(\frac{1}{2}\right)^n.$$ Deduce the limit of the sequence $(u_n)$ and interpret it in the context of the modeling.

In 2020, an influencer on social media has 1000 followers on her profile. The number of followers is modelled as follows: each year, she loses $10\%$ of her followers to which 250 new followers are added. For any natural integer $n$, we denote $u_{n}$ the number of followers on her profile in the year $(2020 + n)$, following this model. Thus $u_{0} = 1000$.

1. Calculate $u_{1}$.
2. Justify that for any natural integer $n$, $u_{n+1} = 0.9 u_{n} + 250$.
3. The Python function named ``suite'' is defined below. In the context of the exercise, interpret the value returned by suite(10).

\begin{verbatim} def suite(n) : u=1000 for i in range(n) : u=0.9*u+250 return u \end{verbatim}

1. a. Show, using a proof by induction, that for any natural integer $n$, $u_{n} \leqslant 2500$. b. Prove that the sequence $(u_{n})$ is increasing. c. Deduce from the previous questions that the sequence $(u_{n})$ is convergent.
2. Let $(v_{n})$ be the sequence defined by $v_{n} = u_{n} - 2500$ for any natural integer $n$. a. Show that the sequence $(v_{n})$ is a geometric sequence with common ratio 0.9 and initial term $v_{0} = -1500$. b. For any natural integer $n$, express $v_{n}$ as a function of $n$ and show that: $$u_{n} = -1500 \times 0.9^{n} + 2500$$ c. Determine the limit of the sequence $(u_{n})$ and interpret it in the context of the exercise.
3. Write a program that determines in which year the number of followers will exceed 2200. Determine this year.

Consider the sequence $\left( u _ { n } \right)$ defined by $u _ { 0 } = 10000$ and for every natural number $n$ :
$$u _ { n + 1 } = 0,95 u _ { n } + 200 .$$

1. Calculate $u _ { 1 }$ and verify that $u _ { 2 } = 9415$.
2. a. Prove, using proof by induction, that for every natural number $n$ : $$u _ { n } > 4000$$ b. It is admitted that the sequence $(u _ { n })$ is decreasing. Justify that it converges.
3. For every natural number $n$, consider the sequence $\left( v _ { n } \right)$ defined by: $v _ { n } = u _ { n } - 4000$. a. Calculate $v _ { 0 }$. b. Prove that the sequence $(v _ { n })$ is geometric with common ratio equal to 0.95. c. Deduce that for every natural number $n$ : $$u _ { n } = 4000 + 6000 \times 0,95 ^ { n }$$ d. What is the limit of the sequence $\left( u _ { n } \right)$? Justify your answer.
4. In 2020, an animal species numbered 10000 individuals. The evolution observed in previous years leads to the estimate that from 2021 onwards, this population will decrease by $5\%$ at the beginning of each year. To slow down this decline, it was decided to reintroduce 200 individuals at the end of each year, starting from 2021.
   A representative of an association supporting this strategy claims that: ``the species should not become extinct, but unfortunately, we will not prevent a loss of more than half the population''. What do you think of this statement? Justify your answer.

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.

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?

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.

At the beginning of the experiment, we have a piece of 2 g of polonium. We know that 1 g of polonium contains $3 \times 10^{21}$ atomic nuclei. We assume that, after 24 hours, $0.5\%$ of the nuclei have disintegrated and that, to compensate for this loss, we then add $0.005\text{ g}$ of polonium. We model the situation using a sequence $\left(v_n\right)_{n \in \mathbb{N}}$; we denote $v_0$ the number of nuclei contained in the polonium at the beginning of the experiment. For $n \geqslant 1$, $v_n$ denotes the number of nuclei contained in the polonium after $n$ days have elapsed.

1. a. Verify that $v_0 = 6 \times 10^{21}$. b. Explain that, for every natural number $n$, we have $$v_{n+1} = 0{,}995\, v_n + 1{,}5 \times 10^{19}.$$
   
2. a. Prove, by induction on $n$, that $0 \leqslant v_{n+1} \leqslant v_n$. b. Deduce that the sequence $\left(v_n\right)_{n \in \mathbb{N}}$ is convergent.
   
3. We consider the sequence $\left(u_n\right)_{n \in \mathbb{N}}$ defined, for every natural number $n$, by: $$u_n = v_n - 3 \times 10^{21}.$$ a. Show that the sequence $\left(u_n\right)_{n \in \mathbb{N}}$ is geometric with common ratio 0.995. b. Deduce that, for every natural number $n$, $v_n = 3 \times 10^{21}\left(0{,}995^n + 1\right)$. c. Deduce the limit of the sequence $\left(v_n\right)_{n \in \mathbb{N}}$ and interpret the result in the context of the exercise.
   
4. Determine, by calculation, after how many days the number of polonium nuclei will be less than $4{,}5 \times 10^{21}$. Justify the answer.
   
5. We wish to have the list of terms of the sequence $\left(v_n\right)_{n \in \mathbb{N}}$. For this, we use a function called \texttt{noyaux} programmed in Python language and partially transcribed below. \begin{verbatim} def noyaux (n) : V =6*10**21 L=[V] for k in range (n) : V=... L.append(V) return L \end{verbatim} a. From reading the previous questions, propose two different solutions to complete line 5 of the \texttt{noyaux} function so that it answers the problem. b. For which value of the integer $n$ will the command \texttt{noyaux(n)} return the daily records of the number of nuclei contained in the polonium sample during 52 weeks of study?

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 $\left( \frac { u _ { n } } { n } \right)$.

For each of the five questions in this exercise, only one of the four proposed answers is correct. No justification is required. A wrong answer, a multiple answer or the absence of an answer to a question neither awards nor deducts points.
We consider L a list of numbers consisting of consecutive terms of an arithmetic sequence with first term 7 and common difference 3, the last number in the list is 2023, namely: $$\mathrm{L} = [7, 10, \ldots, 2023].$$
Question 1: The number of terms in this list is:

Answer A	Answer B	Answer C	Answer D
2023	673	672	2016

Question 2: We choose a number at random from this list. The probability of drawing an even number is:

Answer A	Answer B	Answer C	Answer D
$\frac{1}{2}$	$\frac{34}{673}$	$\frac{336}{673}$	$\frac{337}{673}$

We choose a number at random from this list. We are interested in the following events:

• Event $A$: ``obtain a multiple of 4''
• Event $B$: ``obtain a number whose units digit is 6''

We are given $p(A \cap B) = \frac{34}{673}$.
Question 3: The probability of obtaining a multiple of 4 having 6 as the units digit is:

Answer A	Answer B	Answer C	Answer D
$\frac{168}{673} \times \frac{34}{673}$	$\frac{34}{673}$	$\frac{17}{84}$	$\frac{168}{34}$

Question 4: $P_B(A)$ is equal to:

Answer A	Answer B	Answer C	Answer D
$\frac{36}{168}$	$\frac{1}{2}$	$\frac{33}{168}$	$\frac{34}{67}$

Question 5: We choose, at random, successively, 10 elements from this list. An element can be chosen multiple times. The probability that none of these 10 numbers is a multiple of 4 is:

\begin{tabular}{ c } Answer A

$\left(\frac{505}{673}\right)^{10}$

&

Answer B

$1 - \left(\frac{505}{673}\right)^{10}$

&

Answer C

$\left(\frac{168}{673}\right)^{10}$

&

Answer D

$1 - \left(\frac{168}{673}\right)^{10}$

\hline \end{tabular}

Consider a pyramid with a square base formed of identical balls stacked on top of each other:

• the $1^{\text{st}}$ level, located at the highest level, is composed of 1 ball;
• the $2^{\mathrm{nd}}$ level, just below, is composed of 4 balls;
• the $3^{\mathrm{rd}}$ level has 9 balls;
• the $n$-th level has $n^{2}$ balls.

For any integer $n \geqslant 1$, we denote by $u_{n}$ the number of balls that make up the $n$-th level from the top of the pyramid. Thus, $u_{n} = n^{2}$.

1. Calculate the total number of balls in a pyramid with 4 levels.
2. Consider the sequence $(S_{n})$ defined for any integer $n \geqslant 1$ by $$S_{n} = u_{1} + u_{2} + \ldots + u_{n}.$$ a. Calculate $S_{5}$ and interpret this result. b. Consider the pyramid function below written incompletely in Python. Copy and complete on your paper the box below so that, for any non-zero natural integer $n$, the instruction \texttt{pyramide(n)} returns the number of balls making up a pyramid with $n$ levels. \begin{verbatim} def pyramide(n) : S = 0 for i in range(1, n+1): S = ... return... \end{verbatim} c. Verify that for any natural integer $n$: $$\frac{n(n+1)(2n+1)}{6} + (n+1)^{2} = \frac{(n+1)(n+2)[2(n+1)+1]}{6}$$ d. Prove by induction that for any integer $n \geqslant 1$: $$S_{n} = \frac{n(n+1)(2n+1)}{6}.$$
3. A merchant wishes to arrange oranges in a pyramid with a square base. He has 200 oranges. How many oranges does he use to build the largest possible pyramid?

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 of this exercise are independent.

1. We consider the script written in Python language below. \begin{verbatim} def seuil(S) : n=0 u=7 while u < S : n=n+1 u=1.05*u+3 return(n) \end{verbatim} Statement 1: the instruction seuil(100) returns the value 18.
   
2. Let $(S_n)$ be the sequence defined for every natural integer $n$ by $$S_n = 1 + \frac{1}{5} + \frac{1}{5^2} + \ldots + \frac{1}{5^n}.$$ Statement 2: the sequence $(S_n)$ converges to $\frac{5}{4}$.
   
3. Statement 3: in a class composed of 30 students, we can form 870 different pairs of delegates.
   
4. We consider the function $f$ defined on $[1 ; +\infty[$ by $f(x) = x(\ln x)^2$. Statement 4: the equation $f(x) = 1$ admits a unique solution in the interval $[1 ; +\infty[$.
   
5. Statement 5: $$\int_0^1 x\mathrm{e}^{-x}\,\mathrm{d}x = \frac{\mathrm{e} - 2}{\mathrm{e}}.$$

A patient must take a dose of 2 ml of a medication every hour. We introduce the sequence $\left( u _ { n } \right)$ such that the term $u _ { n }$ represents the quantity of medication, expressed in ml, present in the body immediately after $n$ doses of medication. We have $u _ { 1 } = 2$ and for every strictly positive natural integer $n$: $u _ { n + 1 } = 2 + 0.8 u _ { n }$.
Part A Using this model, a doctor seeks to determine after how many doses of medication the quantity present in the patient's body is strictly greater than 9 mL.

1. Calculate the value $u _ { 2 }$.
2. Show by induction that: $$u _ { n } = 10 - 8 \times 0.8 ^ { n - 1 } \text { for every strictly positive natural integer } n.$$
3. Determine $\lim _ { n \rightarrow + \infty } u _ { n }$ and give an interpretation of this result in the context of the exercise.
4. Let $N$ be a strictly positive natural integer. Does the inequality $u _ { N } \geqslant 10$ have solutions? Interpret the result of this question in the context of the exercise.
5. Determine after how many doses of medication the quantity of medication present in the patient's body is strictly greater than 9 mL. Justify your approach.

Part B Using the same modeling, the doctor is interested in the average quantity of medication present in the patient's body over time. For this purpose, the sequence ( $S _ { n }$ ) is defined for every strictly positive natural integer $n$ by $$S _ { n } = \frac { u _ { 1 } + u _ { 2 } + \cdots + u _ { n } } { n } .$$ We admit that the sequence ( $S _ { n }$ ) is increasing.

1. Calculate $S _ { 2 }$.
2. Show that for every strictly positive natural integer $n$, $$u _ { 1 } + u _ { 2 } + \cdots + u _ { n } = 10 n - 40 + 40 \times 0.8 ^ { n } .$$
3. Calculate $\lim _ { n \rightarrow + \infty } S _ { n }$.
4. The following mystery function is given, written in Python language: \begin{verbatim} def mystere(k) : n = 1 s =2 while sJustify that this value is strictly greater than 10.

A patient must take a dose of 2 mL of a medication every hour.\nWe introduce the sequence $\left( u _ { n } \right)$ such that the term $u _ { n }$ represents the quantity of medication, expressed in mL, present in the body immediately after $n$ doses of medication.\nWe have $u _ { 1 } = 2$ and for every strictly positive natural number $n$: $u _ { n + 1 } = 2 + 0.8 u _ { n }$.

Part A

Using this model, a doctor seeks to determine after how many doses of medication the quantity present in the patient's body is strictly greater than 9 mL.

1. Calculate the value $u _ { 2 }$.
2. Show, by induction on $n$, that $u _ { n } = 10 - 8 \times 0.8 ^ { n - 1 }$ for every strictly positive natural number $n$.
3. Determine $\lim _ { n \rightarrow + \infty } u _ { n }$ and give an interpretation of this result in the context of the exercise.
4. Let $N$ be a strictly positive natural number, does the inequality $u _ { N } \geq 10$ have solutions? Interpret the result of this question in the context of the exercise.
5. Determine from how many doses of medication the quantity of medication present in the patient's body is strictly greater than 9 mL. Justify your approach.

Part B

Using the same modeling, the doctor is interested in the average quantity of medication present in the patient's body over time.\nFor this purpose, we define the sequence ( $S _ { n }$ ) defined for every strictly positive natural number $n$ by
$$S _ { n } = \frac { u _ { 1 } + u _ { 2 } + \cdots + u _ { n } } { n }$$
We admit that the sequence ( $S _ { n }$ ) is increasing.

1. Calculate $S _ { 2 }$.
2. Show that, for every strictly positive natural number $n$,

$$u _ { 1 } + u _ { 2 } + \cdots + u _ { n } = 10 n - 40 + 40 \times 0.8 ^ { n }$$

1. Calculate $\lim _ { n \rightarrow + \infty } S _ { n }$.
2. The following mystery function is given, written in Python language.

\begin{verbatim} def mystere(k): $\mathrm { n } = 1$ $\mathrm { s } = 2$ while $\mathrm { s } < \mathrm { k }$ : $\mathrm { n } = \mathrm { n } + 1$ $\mathrm { s } = 10 - 40 / \mathrm { n } + ( 40 * 0.8 * * \mathrm { n } ) / \mathrm { n }$ return $n$ \end{verbatim}
In the context of the statement, what does the value returned by the input mystere (9) represent?\nDetermine the value returned by the input mystere (9).

1. Justify that this value is strictly greater than 10.

Question 149
A sequência $(a_n)$ é definida por $a_1 = 2$ e $a_{n+1} = 3a_n - 1$, para todo $n \geq 1$. O valor de $a_4$ é
(A) 14 (B) 20 (C) 38 (D) 41 (E) 56

Question 159
A progressão aritmética $(a_n)$ tem primeiro termo $a_1 = 3$ e razão $r = 4$. O valor de $a_{10}$ é
(A) 35 (B) 39 (C) 40 (D) 43 (E) 47

A sequência $(a_n)$ é uma progressão aritmética com $a_1 = 3$ e razão $r = 4$. O valor de $a_{10}$ é
(A) 35 (B) 39 (C) 40 (D) 43 (E) 47