\begin{exercice}
Un carré a pour côté 6.

On considère les étapes suivantes d'une construction où les sommets des triangles se situent au milieu des segments et les triangles tracés sont rectangles.

La construction peut alors se poursuivre à l'infini.
\setlength{\columnseprule}{0pt}
\vspace{-0.75\baselineskip}\begin{multicols}{2}
\begin{center}

\vspace{-0.75\baselineskip}\'Etape 1


\vspace{-0.875\baselineskip}\begin{tikzpicture}[general,scale=0.45]
\clip(-1,-1) rectangle (5,5);
\fill[fill=black,pattern=north east lines] (0,0) -- (4,0) -- (0,4) -- cycle;
\draw (0,0)-- (4,0);
\draw (4,0)-- (4,4);
\draw (4,4)-- (0,4);
\draw (0,4)-- (0,0);
\draw (4,0)-- (0,4);
\end{tikzpicture}

\vspace{-\baselineskip}\'Etape 2


\vspace{-0.875\baselineskip}\begin{tikzpicture}[general,scale=0.45]
\clip(-1,-1) rectangle (5,5);
\fill[fill=black,pattern=north east lines] (0,0) -- (4,0) -- (0,4) -- cycle;
\fill[fill=black,pattern=north east lines] (2,4) -- (2,2) -- (4,2) -- cycle;
\draw (0,0)-- (4,0);
\draw (4,0)-- (4,4);
\draw (4,4)-- (0,4);
\draw (0,4)-- (0,0);
\draw (4,0)-- (0,4);
\draw (2,4)-- (2,2);
\draw (2,2)-- (4,2);
\draw (4,2)-- (2,4);
\end{tikzpicture}



\vspace{-0.75\baselineskip}\'Etape 3


\vspace{-0.875\baselineskip}\begin{tikzpicture}[general,scale=0.45]
\clip(-1,-1) rectangle (5,5);
\fill[fill=black,pattern=north east lines] (0,0) -- (4,0) -- (0,4) -- cycle;
\fill[fill=black,pattern=north east lines] (2,4) -- (2,2) -- (4,2) -- cycle;
\fill[fill=black,pattern=north east lines] (3,4) -- (3,3) -- (4,3) -- cycle;
\draw (0,0)-- (4,0);
\draw (4,0)-- (4,4);
\draw (4,4)-- (0,4);
\draw (0,4)-- (0,0);
\draw (4,0)-- (0,4);
\draw (2,4)-- (2,2);
\draw (2,2)-- (4,2);
\draw (4,2)-- (2,4);
\draw (3,4)-- (3,3);
\draw (3,3)-- (4,3);
\draw (4,3)-- (3,4);
\end{tikzpicture}

\vspace{-\baselineskip}\'Etape 4


\vspace{-0.875\baselineskip}\begin{tikzpicture}[general,scale=0.45]
\clip(-1,-1) rectangle (5,5);
\fill[fill=black,pattern=north east lines] (0,0) -- (4,0) -- (0,4) -- cycle;
\fill[fill=black,pattern=north east lines] (2,4) -- (2,2) -- (4,2) -- cycle;
\fill[fill=black,pattern=north east lines] (3,4) -- (3,3) -- (4,3) -- cycle;
\fill[fill=black,pattern=north east lines] (3.5,4) -- (3.5,3.5) -- (4,3.5) -- cycle;
\draw (0,0)-- (4,0);
\draw (4,0)-- (4,4);
\draw (4,4)-- (0,4);
\draw (0,4)-- (0,0);
\draw (4,0)-- (0,4);
\draw (2,4)-- (2,2);
\draw (2,2)-- (4,2);
\draw (4,2)-- (2,4);
\draw (3,4)-- (3,3);
\draw (3,3)-- (4,3);
\draw (4,3)-- (3,4);
\draw (3.5,4)-- (3.5,3.5);
\draw (3.5,3.5)-- (4,3.5);
\draw (4,3.5)-- (3.5,4);
\end{tikzpicture}
\end{center}
\end{multicols}
\vspace{-2\baselineskip}On note $u_n$ l'aire du triangle ajouté à
l'étape $n$ et $A_n$ l'aire totale hachurée à l'étape $n$ pour $n\geqslant1$.
\begin{enumerate}
\item Déterminer $A_1$ et $A_2$.
\item Exprimer $u_{n+1}$ en fonction de $u_n$.
\item Déterminer $\displaystyle \lim_{n\rightarrow +\infty} A_n$.
\end{enumerate}
\end{exercice}

