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}

Part A

Let $\left(u_{n}\right)$ be the sequence defined by $u_{0} = 30$ and, for every natural integer $n$, $u_{n+1} = \frac{1}{2} u_{n} + 10$. Let $(v_{n})$ be the sequence defined for every natural integer $n$ by $v_{n} = u_{n} - 20$.

1. Calculate the exact values of $u_{1}$ and $u_{2}$.
2. Prove that the sequence $(v_{n})$ is geometric with common ratio $\frac{1}{2}$.
3. Express $v_{n}$ as a function of $n$ for every natural integer $n$.
4. Deduce that, for every natural integer $n$, $u_{n} = 20 + 10\left(\frac{1}{2}\right)^{n}$.
5. Determine the limit of the sequence $\left(u_{n}\right)$. Justify the answer.

Part B

Let $(w_{n})$ be the sequence defined for every natural integer $n$ by: $$\left\{\begin{array}{l} w_{0} = 45 \\ w_{n+1} = \frac{1}{2} w_{n} + \frac{1}{2} u_{n} + 7 \end{array}\right.$$

1. Show that $w_{1} = 44.5$.

We wish to write a function \texttt{suite}, in Python language, which returns the value of the term $w_{n}$ for a given value of $n$. We give below a proposal for this function \texttt{suite}. \begin{verbatim} def suite(n) : U=30 W=45 for i in range (1,n+1) : U=U/2+10 W=W/2+U/2+7 return W \end{verbatim}

1. The execution of \texttt{suite(1)} does not return the term $w_{1}$. How should the function \texttt{suite} be modified so that the execution of \texttt{suite(n)} returns the value of the term $w_{n}$?
2. a. Show, by induction on $n$, that for every natural integer $n$ we have: $$w_{n} = 10n\left(\frac{1}{2}\right)^{n} + 11\left(\frac{1}{2}\right)^{n} + 34$$ b. We admit that for every natural integer $n \geqslant 4$, we have: $0 \leqslant 10n\left(\frac{1}{2}\right)^{n} \leqslant \frac{10}{n}$. What can we deduce about the convergence of the sequence $\left(w_{n}\right)$?

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}

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.

Consider the function $f$ defined for all real $x$ by: $$f ( x ) = \ln \left( \mathrm { e } ^ { \frac { x } { 2 } } + 2 \right)$$ It is admitted that the function $f$ is differentiable on $\mathbb { R }$. Consider the sequence $(u_n)$ defined by $u _ { 0 } = \ln ( 9 )$ and, for all natural integer $n$, $$u _ { n + 1 } = f \left( u _ { n } \right)$$

1. Show that the function $f$ is strictly increasing on $\mathbb { R }$.
2. Show that $f ( 2 \ln ( 2 ) ) = 2 \ln ( 2 )$.
3. Show that $u _ { 1 } = \ln ( 5 )$.
4. Show by induction that for all natural integer $n$, we have: $$2 \ln ( 2 ) \leqslant u _ { n + 1 } \leqslant u _ { n }$$
5. Deduce that the sequence $(u_n)$ converges.
6. a. Solve in $\mathbb { R }$ the equation $X ^ { 2 } - X - 2 = 0$. b. Deduce the set of solutions on $\mathbb { R }$ of the equation: $$\mathrm { e } ^ { x } - \mathrm { e } ^ { \frac { x } { 2 } } - 2 = 0$$ c. Deduce the set of solutions on $\mathbb { R }$ of the equation $f ( x ) = x$. d. Determine the limit of the sequence $\left( u _ { n } \right)$.

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 $]1{,}99;2{,}01[$.

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

With the objective of working on concentration and movement synchronization of students in one of his classes, a physical education teacher divided this class into three groups (A, B, and C) and stipulated the following activity: students in group A should clap every 2 s, students in group B should clap every 3 s, and students in group C should clap every 4 s.
The teacher reset the stopwatch and the three groups started clapping when it registered 1 s. The movements continued until the stopwatch registered 60 s.
An intern noted on paper the sequence formed by the instants when the three groups clapped simultaneously.
What is the general term of the sequence noted?
(A) $12n$, with $n$ a natural number, such that $1 \leq n \leq 5$.
(B) $24n$, with $n$ a natural number, such that $1 \leq n \leq 2$.
(C) $12(n-1)$, with $n$ a natural number, such that $1 \leq n \leq 6$.
(D) $12(n-1)+1$, with $n$ a natural number, such that $1 \leq n \leq 5$.
(E) $24(n-1)+1$, with $n$ a natural number, such that $1 \leq n \leq 3$.

Let $x _ { n }$ be a sequence with the following property: Every subsequence of $x _ { n }$ has a further subsequence which converges to $x$. Then the sequence $x _ { n }$ converges to $x$.

Let $f : ( 0 , \infty ) \longrightarrow \mathbb { R }$ be a continuous function. Then $f$ maps any Cauchy sequence to a Cauchy sequence.

Let $\left\{ f _ { n } : \mathbb { R } \longrightarrow \mathbb { R } \right\}$ be a sequence of continuous functions. Let $x _ { n } \longrightarrow x$ be a convergent sequence of reals. If $f _ { n } \longrightarrow f$ uniformly then $f _ { n } \left( x _ { n } \right) \longrightarrow f ( x )$.

Let $K \subset \mathbb { R } ^ { n }$ such that every real valued continuous function on $K$ is bounded. Then $K$ is compact (i.e closed and bounded).

If $A \subset \mathbb { R } ^ { 2 }$ is a countable set, then $\mathbb { R } ^ { 2 } \backslash A$ is connected.

The set $A = \left\{ ( z , w ) \in \mathbb { C } ^ { 2 } \mid z ^ { 2 } + w ^ { 2 } = 1 \right\}$ is bounded in $\mathbb { C } ^ { 2 }$.

Let $f , g : \mathbb { C } \longrightarrow \mathbb { C }$ be complex analytic, and let $h : [ 0,1 ] \longrightarrow \mathbb { C }$ be a non-constant continuous map. Suppose $f ( z ) = g ( z )$ for every $z \in \operatorname { Im } h$, then $f = g$. (Here $\operatorname { Im } h$ denotes the image of the function $h$.)

There is a non-constant continuous function $f : \mathbb { R } \rightarrow \mathbb { R }$ whose image is contained in $\mathbb { Q }$.

Suppose $f : \mathbb { R } \mapsto \mathbb { R } ^ { n }$ be a differentiable mapping satisfying $\| f ( t ) \| = 1$ for all $t \in \mathbb { R }$. Show that $\left\langle f ^ { \prime } ( t ) , f ( t ) \right\rangle = 0$ for all $t \in \mathbb { R }$. (Here $\|$.$\|$ denotes standard norm or length of a vector in $\mathbb { R } ^ { n }$, and $\langle . , .\rangle$ denotes the standard inner product (or scalar product) in $\mathbb { R } ^ { n }$.)

Let $A , B \subset \mathbb { R } ^ { n }$ and define $A + B = \{ a + b ; a \in A , b \in B \}$. If $A$ and $B$ are open, is $A + B$ open? If $A$ and $B$ are closed, is $A + B$ closed? Justify your answers.

Let $f : X \mapsto Y$ be continuous map onto $Y$, and let $X$ be compact. Also $g : Y \mapsto Z$ is such that $g \circ f$ is continuous. Show $g$ is continuous.

Let $A$ be a $n \times m$ matrix with real entries, and let $B = A A ^ { t }$ and let $\alpha$ be the supremum of $x ^ { t } B x$ where supremum is taken over all vectors $x \in \mathbb { R } ^ { n }$ with norm less than or equal to 1. Consider $$C _ { k } = I + \sum _ { j = 1 } ^ { k } B ^ { j }$$ Show that the sequence of matrices $C _ { k }$ converges if and only if $\alpha < 1$.

Show that a power series $\sum _ { n \geq 0 } a _ { n } z ^ { n }$ where $a _ { n } \rightarrow 0$ as $n \rightarrow \infty$ cannot have a pole on the unit circle. Is the statement true with the hypothesis that $\left( a _ { n } \right)$ is a bounded sequence?

Show that a biholomorphic map of the unit ball onto itself which fixes the origin is necessarily a rotation.

Show that $$\lim _ { x \rightarrow \infty } \frac { x ^ { 100 } \ln ( x ) } { e ^ { x } \tan ^ { - 1 } \left( \frac { \pi } { 3 } + \sin x \right) } = 0$$

Let $x_n = \left(1 - \frac{1}{n}\right) \sin \frac{n\pi}{3}$, $n \geq 1$. Write $l = \liminf x_n$ and $s = \limsup x_n$. Choose the correct statement(s) from below:
(A) $-\frac{\sqrt{3}}{2} \leq l < s \leq \frac{\sqrt{3}}{2}$;
(B) $-\frac{1}{2} \leq l < s \leq \frac{1}{2}$;
(C) $l = -1$ and $s = 1$;
(D) $l = s = 0$.