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%\usepackage{opensans}          % Can't make this work so far. Shame. Would be lovely.
\usepackage[normalem]{ulem}     % For underlining links
% Most of the following are not required for the majority
% of cheat sheets but are needed for some symbol support.
\usepackage{amsmath}            % Symbols
\usepackage{MnSymbol}           % Symbols
\usepackage{wasysym}            % Symbols
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% Document Info
\author{vvvalentined}
\pdfinfo{
  /Title (les-suites.pdf)
  /Creator (Cheatography)
  /Author (vvvalentined)
  /Subject (Les suites Cheat Sheet)
}

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% Header and Footer
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\fancyhead{} % Set header to blank
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\fancyhead[L]{
\noindent
\begin{multicols}{3}
\begin{tabulary}{5.8cm}{C}
    \SetRowColor{DarkBackground}
    \vspace{-7pt}
    {\parbox{\dimexpr\textwidth-2\fboxsep\relax}{\noindent
        \hspace*{-6pt}\includegraphics[width=5.8cm]{/web/www.cheatography.com/public/images/cheatography_logo.pdf}}
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\columnbreak
\begin{tabulary}{11cm}{L}
    \vspace{-2pt}\large{\bf{\textcolor{DarkBackground}{\textrm{Les suites Cheat Sheet}}}} \\
    \normalsize{by \textcolor{DarkBackground}{vvvalentined} via \textcolor{DarkBackground}{\uline{cheatography.com/180334/cs/37524/}}}
\end{tabulary}
\end{multicols}}

\fancyfoot[L]{ \footnotesize
\noindent
\begin{multicols}{3}
\begin{tabulary}{5.8cm}{LL}
  \SetRowColor{FootBackground}
  \mymulticolumn{2}{p{5.377cm}}{\bf\textcolor{white}{Cheatographer}}  \\
  \vspace{-2pt}vvvalentined \\
  \uline{cheatography.com/vvvalentined} \\
  \end{tabulary}
\vfill
\columnbreak
\begin{tabulary}{5.8cm}{L}
  \SetRowColor{FootBackground}
  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}}  \\
   \vspace{-2pt}Not Yet Published.\\
   Updated 5th March, 2023.\\
   Page {\thepage} of \pageref{LastPage}.
\end{tabulary}
\vfill
\columnbreak
\begin{tabulary}{5.8cm}{L}
  \SetRowColor{FootBackground}
  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Sponsor}}  \\
  \SetRowColor{white}
  \vspace{-5pt}
  %\includegraphics[width=48px,height=48px]{dave.jpeg}
  Measure your website readability!\\
  www.readability-score.com
\end{tabulary}
\end{multicols}}




\begin{document}
\raggedright
\raggedcolumns

% Set font size to small. Switch to any value
% from this page to resize cheat sheet text:
% www.emerson.emory.edu/services/latex/latex_169.html
\footnotesize % Small font.

\begin{multicols*}{3}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Raisonnement par récurrence}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{1ère étape : Initialisation}} On vérifie que la propriété est vraie au rang initial. (n=0 ou n=1 en général)} \tn 
% Row Count 3 (+ 3)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{{\bf{2ème étape : Hérédité}} On suppose que la propriété est vraie pour un rang n fixé de ℕ et on démontre qu'elle est vraie au rang n+1.} \tn 
% Row Count 6 (+ 3)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{3ème étape : Conclusion}} On en déduit que la propriété est vraie pour tout n∈ℕ.} \tn 
% Row Count 8 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Vocabulaire et méthodes sur les suites}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Etudier la monotonie d'une suite, c'est étudier son sens de variation.} \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{On peut utiliser un raisonnement par récurrence.} \tn 
% Row Count 3 (+ 1)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{On étudie le signe de Un+1=Un.} \tn 
% Row Count 4 (+ 1)
% Row 3
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{Si Un=f(n), alors on peut étudier le sens de variation de f sur {[}0,+∞{[}.} \tn 
% Row Count 6 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Suites arithmétiques}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Si (Un) est arithmétique, chaque terme permet de déduire le suivant en lui ajoutant une constante appelée raison.} \tn 
% Row Count 3 (+ 3)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{Formule de récurrence: Un+1 = Un+r} \tn 
% Row Count 4 (+ 1)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Formules explicites: Un = U0+r×n ou Un = U1+r×(n-1) {\emph{(si U1 est le premier terme)}}} \tn 
% Row Count 6 (+ 2)
% Row 3
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{Somme des termes consécutifs: (nombre de \seqsplit{termes×(premier+dernier))/2}} \tn 
% Row Count 8 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Suites géométriques}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Si (Un) est géométrique, chaque terme permet de déduire le suivant par multiplication par un facteur constant appelé raison} \tn 
% Row Count 3 (+ 3)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{Formule de récurrence: Un+1 = Un×q} \tn 
% Row Count 4 (+ 1)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Formules explicites: Un = U0×q\textasciicircum{}n\textasciicircum{} ou Un = U0×q\textasciicircum{}n-1\textasciicircum{} (si U1 est le premier terme)} \tn 
% Row Count 6 (+ 2)
% Row 3
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{Somme des termes consécutifs: premier terme × (1-q\textasciicircum{}nombre de termes\textasciicircum{})/1-q} \tn 
% Row Count 8 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Exemple 1}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Soit (Un) la suite définie par U0=-1 et Un+1=Un+2n+2. Démontrer par récurrence que, pour tout n∈ℕ, Un=n$^{\textrm{2}}$+n-1} \tn 
% Row Count 3 (+ 3)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{On vérifie que la propriété est vraie au rang initial. Pour n=0: 0$^{\textrm{2}}$+0-1=-1 et U0=-1} \tn 
% Row Count 5 (+ 2)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{On suppose que la propriété est vraie pour un rang n fixé de ℕ : Un=n$^{\textrm{2}}$+n-1 et on démontre qu'elle est vraie au rang n+1 : Un+1=(n+1)$^{\textrm{2}}$+(n+1)-1} \tn 
% Row Count 8 (+ 3)
% Row 3
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{On sait que Un+1= Un+2n-2 = n$^{\textrm{2}}$+n-1+2n+2 = n$^{\textrm{2}}$+3n+1} \tn 
% Row Count 10 (+ 2)
% Row 4
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{On développe : (n+1)$^{\textrm{2}}$+(n+1)-1 = n$^{\textrm{2}}$+2n+1+n+1-1 = n$^{\textrm{2}}$+3n+1} \tn 
% Row Count 12 (+ 2)
% Row 5
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{On en conclut que, pour tout n∈ℕ, Un=n$^{\textrm{2}}$+n-1} \tn 
% Row Count 13 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Exemple 2}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{(Un) est définie sur ℕ par U0=90 et U1=0.85Un+6. Démontrer par récurrence que (Un) est décroissante sur ℕ.} \tn 
% Row Count 3 (+ 3)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{On vérifie que la propriété est vraie au rang initial. On calcule U1 = 0.85×U0+6 = 82.5 donc U1\textless{}U0} \tn 
% Row Count 6 (+ 3)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{On suppose que la propriété Un+1 \textless{}= Un est vraie pour un rang n fixé de ℕ et on montre que Un+2\textless{}=Un+1:} \tn 
% Row Count 9 (+ 3)
% Row 3
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{Un+1\textless{}=Un+2 (hypothèse de récurrence) ⇔ 0.85Un+1\textless{}=0.85Un ⇔ 0.85Un+1+6\textless{}=0.85Un+6 ⇔ Un+2\textless{}=Un+1} \tn 
% Row Count 11 (+ 2)
% Row 4
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{On en conclut que le suite (Un) est décroissante sur ℕ.} \tn 
% Row Count 13 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}


% That's all folks
\end{multicols*}

\end{document}