Consider the sequence $(u_n)$ defined by:
$$\left\{ \begin{aligned} u _ { 0 } & = 1 \text{ and, for every natural number } n, \\ u _ { n + 1 } & = \left( \frac { n + 1 } { 2 n + 4 } \right) u _ { n } . \end{aligned} \right.$$
Define the sequence $(v_n)$ by: for every natural number $n$, $v_n = (n + 1) u_n$.

1. The spreadsheet below presents the values of the first terms of the sequences $(u_n)$ and $(v_n)$, rounded to the hundred-thousandth. What formula, then extended downward, can be written in cell B3 of the spreadsheet to obtain the successive terms of $(u_n)$?
   

   	A	B	C
   1	$n$	$u_n$	$v_n$
   2	0	1.00000	1.00000
   3	1	0.25000	0.50000
   4	2	0.08333	0.25000
   5	3	0.03125	0.12500
   6	4	0.01250	0.06250
   7	5	0.00521	0.03125
   8	6	0.00223	0.01563
   9	7	0.00098	0.00781
   10	8	0.00043	0.00391
   11	9	0.00020	0.00195

   
   
2. a. Conjecture the expression of $v_n$ as a function of $n$. b. Prove this conjecture.
3. Determine the limit of the sequence $(u_n)$.

For $n \geqslant 1$, we set $p_n = P(A_n)$ with $p_1 = 1$ and $p_{n+1} = 0{,}5\, p_n + 0{,}4$. We set for all integer $n \geqslant 1$: $v_n = p_n - 0{,}8$. a. Prove that $(v_n)$ is a geometric sequence and give its first term $v_1$ and common ratio. b. Express $v_n$ as a function of $n$. Deduce that, for all $n \geqslant 1$, $p_n = 0{,}8 + 0{,}2 \times 0{,}5^{n-1}$. c. Determine the limit of the sequence $(p_n)$.

(Candidates who have not followed the specialisation course)
A computer game of chance is set up as follows:

• If the player wins a game, the probability that he wins the next game is $\frac{1}{4}$;
• If the player loses a game, the probability that he loses the next game is $\frac{1}{2}$;
• The probability of winning the first game is $\frac{1}{4}$.

For every non-zero natural number $n$, we denote by $G_{n}$ the event ``the $n^{\mathrm{th}}$ game is won'' and we denote by $p_{n}$ the probability of this event. We thus have $p_{1} = \frac{1}{4}$.

1. Show that $p_{2} = \frac{7}{16}$.
2. Show that, for every non-zero natural number $n$, $p_{n+1} = -\frac{1}{4} p_{n} + \frac{1}{2}$.
3. We thus obtain the first values of $p_{n}$:
   

   $n$	1	2	3	4	5	6	7
   $p_{n}$	0,25	0,4375	0,3906	0,4023	0,3994	0,4001	0,3999

   
   What conjecture can be made?
4. We define, for every non-zero natural number $n$, the sequence $(u_{n})$ by $u_{n} = p_{n} - \frac{2}{5}$. a. Prove that the sequence $(u_{n})$ is a geometric sequence and specify its common ratio. b. Deduce that, for every non-zero natural number $n$, $p_{n} = \frac{2}{5} - \frac{3}{20}\left(-\frac{1}{4}\right)^{n-1}$. c. Does the sequence $(p_{n})$ converge? Interpret this result.

Newton's law of cooling states that the rate of change of the temperature of a body is proportional to the difference between the temperature of this body and that of the surrounding environment.
A cup of coffee is served at an initial temperature of $80^{\circ}\mathrm{C}$ in an environment whose temperature, expressed in degrees Celsius, assumed to be constant, is denoted $M$.
In this part, for any natural number $n$, we denote $T_n$ the temperature of the coffee at instant $n$, with $T_n$ expressed in degrees Celsius and $n$ in minutes. Thus $T_0 = 80$.
We model Newton's law between any two consecutive minutes $n$ and $n+1$ by the equality: $$T_{n+1} - T_n = k(T_n - M)$$ where $k$ is a real constant.
In the rest of part A, we choose $M = 10$ and $k = -0{,}2$. Thus, for any natural number $n$, we have: $T_{n+1} - T_n = -0{,}2(T_n - 10)$.

1. Based on the context, can we conjecture the direction of variation of the sequence $(T_n)$?
2. Show that for any natural number $n$: $T_{n+1} = 0{,}8T_n + 2$.
3. We set, for any natural number $n$: $u_n = T_n - 10$. a. Show that $(u_n)$ is a geometric sequence. Specify its common ratio and its first term $u_0$. b. Show that, for any natural number $n$, we have: $T_n = 70 \times 0{,}8^n + 10$. c. Determine the limit of the sequence $(T_n)$.
4. Consider the following algorithm: \begin{verbatim} While $T \geqslant 40$ $T \leftarrow 0,8T + 2$ $n \leftarrow n + 1$ End While \end{verbatim} a. Initially, we assign the value 80 to the variable $T$ and the value 0 to the variable $n$. What numerical value does the variable $n$ contain at the end of the algorithm's execution? b. Interpret this value in the context of the exercise.

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.

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.

In this exercise, we consider the sequence ( $T _ { n }$ ) defined by:
$$T _ { 0 } = 180 \mathrm { and } , \text { for all natural integer } n , T _ { n + 1 } = 0,955 T _ { n } + 0,9$$

1. a. Prove by induction that, for all natural integer $n , T _ { n } \geqslant 20$. b. Verify that for all natural integer $n , T _ { n + 1 } - T _ { n } = - 0,045 \left( T _ { n } - 20 \right)$. Deduce the direction of variation of the sequence ( $T _ { n }$ ). c. Conclude from the above that the sequence ( $T _ { n }$ ) is convergent. Justify.
   
2. For all natural integer $n$, we set: $u _ { n } = T _ { n } - 20$. a. Show that the sequence ( $u _ { n }$ ) is a geometric sequence and specify its common ratio. b. Deduce that for all natural integer $n , T _ { n } = 20 + 160 \times 0,955 ^ { n }$. c. Calculate the limit of the sequence ( $T _ { n }$ ). d. Solve the inequality $T _ { n } \leqslant 120$ with unknown $n$ a natural integer.
   
3. In this part, we are interested in the evolution of temperature at the center of a cake after it comes out of the oven. We consider that when the cake comes out of the oven, the temperature at the center of the cake is $180 ^ { \circ } \mathrm { C }$ and that of the ambient air is $20 ^ { \circ } \mathrm { C }$. Newton's law of cooling allows us to model the temperature at the center of the cake by the previous sequence ( $T _ { n }$ ). More precisely, $T _ { n }$ represents the temperature at the center of the cake, expressed in degrees Celsius, $n$ minutes after it comes out of the oven. a. Explain why the limit of the sequence ( $T _ { n }$ ) determined in question 2. c. was foreseeable in the context of the exercise. b. We consider the following Python function:

\begin{verbatim} def temp(x) : T = 180 n = 0 while T > x : T=0.955*T+0.9 n=n+1 return n \end{verbatim}
Give the result obtained by executing the command temp(120). Interpret the result in the context of the exercise.

Exercise 2 Sequences
Let $\left(u_{n}\right)$ be the sequence defined by $u_{0} = 4$ and, for every natural integer $n$, $u_{n+1} = \frac{1}{5} u_{n}^{2}$.

1. a. Calculate $u_{1}$ and $u_{2}$. b. Copy and complete the function below written in Python language. This function is named suite\_u and takes as parameter the natural integer $p$. It returns the value of the term of rank $p$ of the sequence $(u_{n})$. \begin{verbatim} def suite_u(p) : u= ... for i in range(1,...) : u =... return u \end{verbatim}
2. a. Prove by induction that for every natural integer $n$, $0 < u_{n} \leqslant 4$. b. Prove that the sequence $(u_{n})$ is decreasing. c. Deduce from this that the sequence $(u_{n})$ is convergent.
3. a. Justify that the limit $\ell$ of the sequence $(u_{n})$ satisfies the equality $\ell = \frac{1}{5} \ell^{2}$. b. Deduce from this the value of $\ell$.
4. For every natural integer $n$, we set $v_{n} = \ln\left(u_{n}\right)$ and $w_{n} = v_{n} - \ln(5)$. a. Show that, for every natural integer $n$, $v_{n+1} = 2v_{n} - \ln(5)$. b. Show that the sequence $(w_{n})$ is geometric with common ratio 2. c. For every natural integer $n$, give the expression of $w_{n}$ as a function of $n$ and show that $v_{n} = \ln\left(\frac{4}{5}\right) \times 2^{n} + \ln(5)$.
5. Calculate $\lim_{n \rightarrow +\infty} v_{n}$ and find again $\lim_{n \rightarrow +\infty} u_{n}$.

In a large French city, electric scooters are made available to users. A company, responsible for maintaining the scooter fleet, checks their condition every Monday.
It is estimated that:

• when a scooter is in good condition on a Monday, the probability that it is still in good condition the following Monday is 0.9;
• when a scooter is in poor condition on a Monday, the probability that it is in good condition the following Monday is 0.4.

We are interested in the condition of a scooter during the inspection phases. Let $n$ be a natural integer. We denote $B_n$ the event ``the scooter is in good condition $n$ weeks after its commissioning'' and $p_n$ the probability of $B_n$. When commissioned, the scooter is in good condition. We therefore have $p_0 = 1$.

1. Give $p_1$ and show that $p_2 = 0.85$. You may rely on a weighted tree.
2. Copy and complete the weighted tree.
3. Deduce that, for all natural integer $n$, $p_{n+1} = 0.5p_n + 0.4$.
4. a. Prove by induction that for all natural integer $n$, $p_n \geqslant 0.8$. b. Based on this result, what communication can the company consider to highlight the reliability of the fleet?
5. a. Consider the sequence $(u_n)$ defined for all natural integer $n$ by $u_n = p_n - 0.8$. Show that $(u_n)$ is a geometric sequence and give its first term and common ratio. b. Deduce the expression of $u_n$ then of $p_n$ as a function of $n$. c. Deduce the limit of the sequence $(p_n)$.

Exercise 1 — 5 points Theme: probability, sequences
Parts A and B can be treated independently
Part A
Each day, an athlete must jump over a hurdle at the end of training. Based on the previous season, his coach estimates that

• if the athlete clears the hurdle one day, then he will clear it in $90\%$ of cases the next day;
• if the athlete does not clear the hurdle one day, then in $70\%$ of cases he will not clear it the next day either.

For every natural integer $n$, we denote:

• $R_{n}$ the event: ``The athlete successfully clears the hurdle during the $n$-th session'',
• $p_{n}$ the probability of event $R_{n}$. We consider that $p_{0} = 0.6$.

1. Let $n$ be a natural integer, copy the weighted tree below and complete the blanks.
2. Justify using the tree that, for every natural integer $n$, we have: $$p_{n+1} = 0.6 p_{n} + 0.3 .$$
3. Consider the sequence $(u_{n})$ defined, for every natural integer $n$, by $u_{n} = p_{n} - 0.75$. a. Prove that the sequence $(u_{n})$ is a geometric sequence and specify its common ratio and first term. b. Prove that, for every natural integer $n$: $$p_{n} = 0.75 - 0.15 \times 0.6^{n} .$$ c. Deduce that the sequence $(p_{n})$ is convergent and determine its limit $\ell$. d. Interpret the value of $\ell$ in the context of the exercise.

Part B
After many training sessions, the coach now estimates that the athlete clears each hurdle with a probability of 0.75 and this independently of whether or not he cleared the previous hurdles. We denote $X$ the random variable that gives the number of hurdles cleared by the athlete at the end of a 400 metres hurdles race which has 10 hurdles.

1. Specify the nature and parameters of the probability distribution followed by $X$.
2. Determine, to $10^{-3}$ near, the probability that the athlete clears all 10 hurdles.
3. Calculate $p(X \geqslant 9)$, to $10^{-3}$ near.

Léa spends a good part of her days playing a video game and is interested in the chances of winning her next games.
She estimates that if she has just won a game, she wins the next one in $70\%$ of cases. But if she has just suffered a defeat, according to her, the probability that she wins the next one is 0.2. Furthermore, she thinks she has an equal chance of winning the first game as of losing it.
For all non-zero natural integer $n$, we define the following events:

• $G _ { n }$: ``Léa wins the $n$-th game of the day'';
• $D _ { n }$: ``Léa loses the $n$-th game of the day''.

For all non-zero natural integer $n$, we denote $g _ { n }$ the probability of event $G _ { n }$. We have therefore $g _ { 1 } = 0.5$.

1. What is the value of the conditional probability $p _ { G _ { 1 } } \left( D _ { 2 } \right)$?
2. Copy and complete the probability tree below which models the situation for the first two games of the day.
3. Calculate $g _ { 2 }$.
4. Let $n$ be a non-zero natural integer. a. Copy and complete the probability tree below which models the situation for the $n$-th and $(n+1)$-th games of the day. b. Justify that for all non-zero natural integer $n$, $$g _ { n + 1 } = 0.5 g _ { n } + 0.2 .$$
5. For all non-zero natural integer $n$, we set $v _ { n } = g _ { n } - 0.4$. a. Show that the sequence $( v _ { n } )$ is geometric. We will specify its first term and its common ratio. b. Show that, for all non-zero natural integer $n$: $$g _ { n } = 0.1 \times 0.5 ^ { n - 1 } + 0.4 .$$
6. Study the variations of the sequence $( g _ { n } )$.
7. Give, by justifying, the limit of the sequence $( g _ { n } )$. Interpret the result in the context of the problem.
8. Determine, by calculation, the smallest integer $n$ such that $g _ { n } - 0.4 \leqslant 0.001$.
9. Copy and complete lines 4, 5 and 6 of the following function, written in Python language, so that it returns the smallest rank from which the terms of the sequence $\left( g _ { n } \right)$ are all less than or equal to $0.4 + e$, where $e$ is a strictly positive real number. \begin{verbatim} def seuil(e) : g = 0.5 n = 1 while...: g = 0.5 * g + 0.2 n = ... return (n) \end{verbatim}

A robot is positioned on a horizontal axis and moves several times by one meter on this axis, randomly to the right or to the left. During the first movement, the probability that the robot moves to the right is equal to $\frac{1}{3}$. If it moves to the right, the probability that the robot moves to the right again during the next movement is equal to $\frac{3}{4}$. If it moves to the left, the probability that the robot moves to the left again during the next movement is equal to $\frac{1}{2}$. For every natural integer $n \geqslant 1$, we denote:

• $D_n$ the event: ``the robot moves to the right during the $n$-th movement'';
• $\overline{D_n}$ the complementary event of $D_n$;
• $p_n$ the probability of event $D_n$.

We therefore have $p_1 = \frac{1}{3}$.
Part A: study of the special case where $n = 2$ In this part, the robot performs two successive movements.

1. Reproduce and complete the following weighted tree.
2. Determine the probability that the robot moves to the right twice.
3. Show that $p_2 = \frac{7}{12}$.
4. The robot moved to the left during the second movement. What is the probability that it moved to the right during the first movement?

Part B: study of the sequence $(p_n)$. We wish to estimate the movement of the robot after a large number of steps.

1. Prove that for every natural integer $n \geqslant 1$, we have: $$p_{n+1} = \frac{1}{4} p_n + \frac{1}{2}.$$ You may use a tree to help.
2. a. Show by induction that for every natural integer $n \geqslant 1$, we have: $$p_n \leqslant p_{n+1} < \frac{2}{3}.$$ b. Is the sequence $(p_n)$ convergent? Justify.
3. We consider the sequence $(u_n)$ defined for every natural integer $n \geqslant 1$, by $u_n = p_n - \frac{2}{3}$. a. Show that the sequence $(u_n)$ is geometric and specify its first term and its common ratio. b. Determine the limit of the sequence $(p_n)$ and interpret the result in the context of the exercise.

Part C In this part, we consider another robot that performs ten movements of one meter independent of each other, each movement to the right having a fixed probability equal to $\frac{3}{4}$. What is the probability that it returns to its starting point after the ten movements? Round the result to $10^{-3}$ near.

Let $n$ be a non-zero natural integer. In the context of a random experiment, we consider a sequence of events $A _ { n }$ and we denote $p _ { n }$ the probability of the event $A _ { n }$. For parts $\mathbf { A }$ and $\mathbf { B }$ of the exercise, we consider that:

• If the event $A _ { n }$ is realized then the event $A _ { n + 1 }$ is realized with probability 0.3.
• If the event $A _ { n }$ is not realized then the event $A _ { n + 1 }$ is realized with probability 0.7.

We assume that $p _ { 1 } = 1$.

Part A:

1. Copy and complete the probabilities on the branches of the probability tree below.
2. Show that $p _ { 3 } = 0.58$.
3. Calculate the conditional probability $P _ { A _ { 3 } } \left( A _ { 2 } \right)$, round the result to $10 ^ { - 2 }$ near.

Part B:

In this part, we study the sequence $( p _ { n } )$ with $n \geqslant 1$.

1. Copy and complete the probabilities on the branches of the probability tree below.
2. a. Show that, for all non-zero natural integer $n$: $p _ { n + 1 } = - 0.4 p _ { n } + 0.7$.

We consider the sequence $( u _ { n } )$, defined for all non-zero natural integer $n$ by: $u _ { n } = p _ { n } - 0.5$. b. Show that $( u _ { n } )$ is a geometric sequence for which we will specify the common ratio and the first term. c. Deduce the expression of $u _ { n }$, then of $p _ { n }$ as a function of $n$. d. Determine the limit of the sequence $\left( p _ { n } \right)$.

Part C:

Let $x \in ] 0 ; 1 [$, we assume that $P _ { \overline { A _ { n } } } \left( A _ { n + 1 } \right) = P _ { A _ { n } } \left( \overline { A _ { n + 1 } } \right) = x$. We recall that $p _ { 1 } = 1$.

1. Show that for all non-zero natural integer $n$: $p _ { n + 1 } = ( 1 - 2 x ) p _ { n } + x$.
2. Prove by induction on $n$ that, for all non-zero natural integer $n$: $$p _ { n } = \frac { 1 } { 2 } ( 1 - 2 x ) ^ { n - 1 } + \frac { 1 } { 2 }$$
3. Show that the sequence $\left( p _ { n } \right)$ is convergent and give its limit.

Part B - Second model
After studying a larger collection of data over the last 50 years, another modelling appears more relevant:

• if the El Niño phenomenon is dominant in one year, then the probability that it is still dominant the following year is 0.5
• on the other hand, if the El Niño phenomenon is not dominant in one year, then the probability that it is dominant the following year is 0.3.

We consider that the reference year is 2023. We denote for every natural integer $n$:

• $E _ { n }$ the event ``the El Niño phenomenon is dominant in the year $2023 + n$ '';
• $p _ { n }$ the probability of the event $E _ { n }$.

In 2023, El Niño was not dominant. We thus have $p _ { 0 } = 0$.

1. Let $n$ be a natural integer. Copy and complete the following weighted tree.
2. Justify that $p _ { 1 } = 0.3$.
3. Using the tree, show that, for every natural integer $n$, we have: $$p _ { n + 1 } = 0.2 p _ { n } + 0.3$$
4. a. Conjecture the variations and the possible limit of the sequence $( p _ { n } )$. b. Show by induction that, for every natural integer $n$, we have: $p _ { n } \leqslant \frac { 3 } { 8 }$. c. Determine the direction of variation of the sequence $\left( p _ { n } \right)$. d. Deduce the convergence of the sequence $\left( p _ { n } \right)$.
5. Let $\left( u _ { n } \right)$ be the sequence defined by $u _ { n } = p _ { n } - \frac { 3 } { 8 }$ for every natural integer $n$. a. Show that the sequence $( u _ { n } )$ is geometric with ratio 0.2 and specify its first term. b. Show that, for every natural integer $n$, we have: $$p _ { n } = \frac { 3 } { 8 } \left( 1 - 0.2 ^ { n } \right) .$$ c. Calculate the limit of the sequence $\left( p _ { n } \right)$. d. Interpret this result in the context of the exercise.