%\renewcommand{\theenumi}{\Roman{enumi}}
%\renewcommand{\labelenumiv}{\textbf{\theenumiv.}}

\documentclass{article}%
\usepackage{amssymb}
\usepackage[french]{babel}
\usepackage{geometry}
\usepackage{multicol}
\usepackage{lastpage}
\usepackage[french]{layout}
\usepackage{points}
\usepackage{boxedminipage}
\usepackage{fancybox}
\usepackage{sectsty}
\usepackage{theorem}
\usepackage{amsfonts}
\usepackage{amsmath}
\usepackage{fancyhdr}
\usepackage{afterpage}%
\setcounter{MaxMatrixCols}{30}%
\usepackage{graphicx}
%TCIDATA{OutputFilter=latex2.dll}
%TCIDATA{Version=5.00.0.2552}
%TCIDATA{CSTFile=40 LaTeX article.cst}
%TCIDATA{Created=Sunday, October 05, 2003 15:40:22}
%TCIDATA{LastRevised=Thursday, September 02, 2004 19:01:33}
%TCIDATA{<META NAME="GraphicsSave" CONTENT="32">}
%TCIDATA{<META NAME="SaveForMode" CONTENT="1">}
%TCIDATA{<META NAME="DocumentShell" CONTENT="Articles\SW\DevoirJFil">}
\geometry{
a4paper,
marginpar=2cm,
lmargin=0cm,
rmargin=2cm,
marginparsep=0pt,
includemp=true,
reversemp=true,
headsep=-0.2in
}
\setlength{\theorempostskipamount}{3\baselineskip}
\theoremheaderfont{\large\bfseries}
{  \theorembodyfont{\large\upshape\mdseries}
\theoremstyle{break}
\newtheorem{theorem}{Théorème}
}
\newcounter{Cack}[section]
{   \theorembodyfont{\large\upshape}
\theoremstyle{break}
\newtheorem  {acknowledgement}[Cack]{A savoir}
}
\newcounter{Calg}[section]
{  \theorembodyfont{\upshape}
\theoremstyle{break}
\newtheorem{algorithm}[Calg]{Méthode}
}
\newcounter{Caxi}[section]
{   \theorembodyfont{\itshape}
\newtheorem{axiom}[Caxi]{Axiome}
}
\newcounter{Ccas}[section]
{   \theorembodyfont{\upshape}
\newtheorem{case}[Ccas]{Cas}
}
\newcounter{Ccorr}[section]
\renewcommand{\theCcorr}{\Roman{Ccorr}}
{   \theorembodyfont{\upshape}
\theoremstyle{break}
\newtheorem{claim}[Ccorr]{Correction de l'exercice}
}
\newcounter{Cconc}[theorem]
{   \theorembodyfont{\large\mdseries}
\newtheorem{conclusion}[Cconc]{Conclusion}
}
\newcounter{Ccond}[theorem]
{  \theorembodyfont{\large\mdseries}
\newtheorem{condition}[Ccond]{Hypothèse}
}
\newcounter{Cconj}[section]
{   \theorembodyfont{\upshape}
\newtheorem{conjecture}[Cconj]{Conjecture}
}
\newcounter{Ccor}[section]
{   \theorembodyfont{\upshape}
\newtheorem{corollary}[Ccor]{Corollaire}
}
\newcounter{Ccrit}[section]
{   \theorembodyfont{\upshape}
\newtheorem{criterion}[Ccrit]{Critère}
}
\newcounter{Cdef}[section]
\newcounter{Cnot}[section]
{   \theorembodyfont{\large\upshape}
\theoremstyle{break}
\newtheorem{definition}[Cdef]{Définition}
\newtheorem{notation}[Cnot]{Notation}
}
\newcounter{Cexa}[section]
{   \theorembodyfont{\slshape}
\newtheorem{example}[Cexa]{Exemple}
}
\newcounter{Cexe}[section]
\renewcommand{\theCexe}{\Roman{Cexe}}
{   \theorembodyfont{\upshape}
\theoremstyle{break}
\newtheorem{exercise}[Cexe]{Exercice}
}
\newcounter{Clem}[section]
{   \theorembodyfont{\upshape}
\newtheorem{lemma}[Clem]{Lemme}
}
\newcounter{Cprob}[section]
\renewcommand{\theCprob}
{\Roman{Cprob}.}
{   \theorembodyfont{\upshape}
\theoremstyle{break}
\newtheorem{problem}[Cprob]{Problème}
}
{   \theorembodyfont{\large\mdseries}
\newcounter{Cprop}[section]
\theoremstyle{break}
\newtheorem{proposition}[Cprop]{Propriété}
}
\newcounter{Crem}[section]
{   \theorembodyfont{\itshape}
\newtheorem{remark}[Crem]{Remarque}
}
\newcounter{Csol}[section]
{   \theorembodyfont{\upshape}
\theoremstyle{break}
\newtheorem{solution}[Csol]{Solution}
}
\newcounter{Csum}[section]
{   \theorembodyfont{\upshape}
\newtheorem{summary}[Csum]{En résumé}
}
\newenvironment{proof}[1]
[Démonstration]
{\noindent\textbf{#1:} }
{\ \fbox{cqfd.}
}
\renewcommand{\theenumi}{\arabic{enumi}}
\renewcommand{\theenumii}{\alph{enumii}}
\renewcommand{\theenumiii}{\roman{enumiii}}
\renewcommand{\labelenumi}{\textbf{\theenumi)}}
\renewcommand{\labelenumii}{\textbf{\theenumii)}}
\renewcommand{\labelenumiii}{\textbf{\theenumiii.}}
\newcommand{\LeTitre}{}
\newcommand{\Titre}[1]{\title{#1} \renewcommand
{\LeTitre}{#1}}
\newcommand{\LaClasse}{}
\newcommand
{\Classe}[1]{\renewcommand{\LaClasse} {#1}}
\newcommand{\LaDate}{}
\newcommand
{\Date}[1]{\author{#1}\renewcommand{\LaDate} {#1}}
\newcommand{\Duree}[1]{\date{#1}}
\newcommand{\Letablissement}{}
\newcommand{\Etablissement}[1]{\renewcommand{\Letablissement} {#1}}
\newcommand{\Ladiscipline}{}
\newcommand{\Discipline}[1]{\renewcommand{\Ladiscipline} {#1}}
\renewcommand{\thesection}
{\Alph{section}}
\renewcommand{\thesubsection}
{\thesection\arabic{subsection}}
\renewcommand{\thesubsubsection}
{\thesubsection.\arabic{subsubsection}}
\begin{document}

%%%%%%%%%%b{gedeonInfo}%%%%%%%%%
%%--Document N° 1186767407377--
%%--suite géométrique auxiliaire et forme explicite--
%%%%%%%%%%e{gedeonInfo}%%%%%%%%%


\begin{exercise}
[{\small extrait d'un sujet de BAC}]On consid\`{e}re la suite $\left(
u_{n}\right)  _{n\in%
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
^{\ast}}$ de nombres r\'{e}els, d\'{e}finie par la relation de r\'{e}currence
$u_{n+1}=\frac{4}{10}-\frac{3}{10}u_{n}$ et par la condition initiale
$u_{1}=a$ ($a$ \'{e}tant un r\'{e}el donn\'{e}).

$\left(  v_{n}\right)  _{n\in%
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
^{\ast}}$ est la suite de nombres r\'{e}els d\'{e}finie par $v_{n}=13u_{n}-4.$

\begin{enumerate}
\item D\'{e}montrer que $(v_{n})$ est une suite g\'{e}om\'{e}trique dont on
d\'{e}terminera la raison $k$.

\item Exprimer $v_{n}$ en fonction de $n$ et de $a.$

\item En d\'{e}duire $u_{n}$ en fonction de $n$ et de $a.$
\end{enumerate}
\end{exercise}

\begin{claim}
$\vspace*{-0.35cm}$

\begin{enumerate}
\item On a: $u_{n+1}=\tfrac{4}{10}-\tfrac{3}{10}u_{n}\quad$et$\quad
v_{n}=13u_{n}-4$\marginpar{%
%TCIMACRO{\TeXButton{Barème}{\hspace{1cm}\fbox{\scriptsize{2 pts}}}}%
%BeginExpansion
\hspace{1cm}\fbox{\scriptsize{2 pts}}%
%EndExpansion
}

donc $v_{n+1}=13\left(  \tfrac{4}{10}-\tfrac{3}{10}u_{n}\right)
-4.=\dfrac{12}{10}-\dfrac{39}{10}u_{n}=-\dfrac{3}{10}\left(  13u_{n}-4\right)
.$

On en d\'{e}duit que, $\forall n\in{%
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
}^{\ast}:\dfrac{v_{n+1}}{v_{n}}=-\dfrac{3}{10};$%
\[
\fbox{ie:\ $(v_{n})$\ est g\'{e}om\'{e}trique de raison\ $k=\frac{-3}{10}$}%
\]


\item \label{vnenfoncn}D'apr\`{e}s la forme explicite d'une suite
g\'{e}om\'{e}trique on a: $v_{n}=v_{0}\times k^{n};$\marginpar{%
%TCIMACRO{\TeXButton{Barème}{\hspace{1cm}\fbox{\scriptsize{1 pt}}}}%
%BeginExpansion
\hspace{1cm}\fbox{\scriptsize{1 pt}}%
%EndExpansion
}%
\[
\fbox{$\text{ie:\ }v_{n}=\left(  13a-4\right)  \left(  \tfrac{-3}{10}\right)
^{n}.$}%
\]


\item Du \ref{vnenfoncn}) et de $v_{n}=13u_{n}-4$ on d\'{e}duit que
$13u_{n}-4=\left(  13a-4\right)  \left(  \tfrac{-3}{10}\right)  ^{n}%
.$\marginpar{%
%TCIMACRO{\TeXButton{Barème}{\hspace{1cm}\fbox{\scriptsize{1 pt}}}}%
%BeginExpansion
\hspace{1cm}\fbox{\scriptsize{1 pt}}%
%EndExpansion
}%
\[
\fbox{$\text{D'o\`{u}\ }u_{n}=\tfrac{1}{13}\left(  13a-4\right)  \left(
-\tfrac{3}{10}\right)  ^{n}+\tfrac{4}{13}.$}%
\]

\end{enumerate}
\end{claim}


\end{document}