(For candidates who have not followed the specialization course) Let the numerical sequence $(u _ { n })$ defined on $\mathbf{N}$ by: $$u _ { 0 } = 2 \quad \text { and for every natural number } n , u _ { n + 1 } = \frac { 2 } { 3 } u _ { n } + \frac { 1 } { 3 } n + 1$$

1. 1. [a.] Calculate $u _ { 1 } , u _ { 2 } , u _ { 3 }$ and $u _ { 4 }$. Approximate values to $10 ^ { - 2 }$ may be given.
   2. [b.] Form a conjecture about the monotonicity of this sequence.
2. 1. [a.] Prove that for every natural number $n$, $$u _ { n } \leqslant n + 3$$
   2. [b.] Prove that for every natural number $n$, $$u _ { n + 1 } - u _ { n } = \frac { 1 } { 3 } \left( n + 3 - u _ { n } \right)$$
   3. [c.] Deduce a validation of the previous conjecture.
3. We denote by $\left( v _ { n } \right)$ the sequence defined on $\mathbf { N }$ by $v _ { n } = u _ { n } - n$.
   1. [a.] Prove that the sequence $\left( v _ { n } \right)$ is a geometric sequence with common ratio $\frac { 2 } { 3 }$.
   2. [b.] Deduce that for every natural number $n$, $$u _ { n } = 2 \left( \frac { 2 } { 3 } \right) ^ { n } + n$$
   3. [c.] Determine the limit of the sequence $(u _ { n })$.
4. For every non-zero natural number $n$, we set: $$S _ { n } = \sum _ { k = 0 } ^ { n } u _ { k } = u _ { 0 } + u _ { 1 } + \ldots + u _ { n } \quad \text { and } \quad T _ { n } = \frac { S _ { n } } { n ^ { 2 } } .$$
   1. [a.] Express $S _ { n }$ as a function of $n$.
   2. [b.] Determine the limit of the sequence $(T _ { n })$.

Consider the numerical sequence $(u_n)$ defined on $\mathbb{N}$ by:
$$u _ { 0 } = 2 \quad \text { and for every natural number } n , \quad u _ { n + 1 } = - \frac { 1 } { 2 } u _ { n } ^ { 2 } + 3 u _ { n } - \frac { 3 } { 2 } .$$

Part A: Conjecture

1. Calculate the exact values, given as irreducible fractions, of $u _ { 1 }$ and $u _ { 2 }$.
2. Give an approximate value to $10 ^ { - 5 }$ of the terms $u _ { 3 }$ and $u _ { 4 }$.
3. Conjecture the direction of variation and the convergence of the sequence $(u_n)$.

Part B: Validation of conjectures

Consider the numerical sequence $\left( v _ { n } \right)$ defined for every natural number $n$, by: $v _ { n } = u _ { n } - 3$.

1. Show that, for every natural number $n , v _ { n + 1 } = - \frac { 1 } { 2 } v _ { n } ^ { 2 }$.
2. Prove by induction that, for every natural number $n , - 1 \leqslant v _ { n } \leqslant 0$.
3. a. Prove that, for every natural number $n , v _ { n + 1 } - v _ { n } = - v _ { n } \left( \frac { 1 } { 2 } v _ { n } + 1 \right)$. b. Deduce the direction of variation of the sequence $\left( v _ { n } \right)$.
4. Why can we then affirm that the sequence $\left( v _ { n } \right)$ converges?
5. Let $\ell$ denote the limit of the sequence $(v_n)$. It is admitted that $\ell$ belongs to the interval $[ - 1 ; 0 ]$ and satisfies the equality: $\ell = - \frac { 1 } { 2 } \ell ^ { 2 }$. Determine the value of $\ell$.
6. Are the conjectures made in Part A validated?

Exercise 5

5 points

Candidates who have not followed the specialized course

Let $k$ be a strictly positive real number. We consider the sequence ( $u _ { n }$ ) defined by $u _ { 0 } = 1 , u _ { 1 } = k$ and, for all natural integer $n$ by:
$$u _ { n + 2 } = \frac { u _ { n + 1 } ^ { 2 } } { k u _ { n } }$$
It is admitted that all terms of the sequence ( $u _ { n }$ ) exist and are strictly positive.

1. Express $u _ { 2 } , u _ { 3 }$ and $u _ { 4 }$ as functions of $k$.
2. Using a spreadsheet, the first terms of the sequence ( $u _ { n }$ ) were calculated for two values of $k$. The value of the real number $k$ is entered in cell E 2 .

	A	B	C	D	E		A	B	C	D	E
1	$n$	$u ( n )$				1	$n$	$u ( n )$
2	0	1		$k =$	2.7182818	2	0	1		$k =$	0.9
3	1	2.7182818				3	1	0.9
4	2	2.7182818				4	2	0.9
5	3	1				5	3	1
6	4	0.1353353				6	4	1.2345679
7	5	0.0067319				7	5	1.6935088
8	6	0.000 1234				8	6	2.581 1748
9	7	$8.315 \mathrm { E } - 07$				9	7	4.3712422
10	8	$2.061 \mathrm { E } - 09$				10	8	8.2252633

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 $u _ { 0 } = 3$ ?

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}