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\begin{center}\section*{\textcolor{red}{Spécialité Première : Devoir sur feuille \no 2 }}\end{center}



\begin{multicols}{2}
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\subsection{}%http://pierrelux.net/documents/bts/cours/module_suite/exercices_suites.pdf
On dispose d'un crédit de \numprint{414000} euros pour atteindre dans un désert une nappe souterraine . Le coût du forage est fixé à \numprint{1000} euros pour le premier mètre creusé, \numprint{1200} pour le deuxième, \numprint{1400} pour le troisième et ainsi de suite en augmentant de 200 euros par mètre creusé.\\
On pose $u_0=\numprint{1000}$ , $u_1=\numprint{1200}$ \dots \\
$u_n$ désigne donc le coût en euros du (n+1)-ième mètre creusé.

\begin{enumerate}
\item 
\begin{enumerate}
\item Calculer $u_5$.

\item Exprimer $u_{n+1}$ en fonction de $u_n$ , pour tout $n\in\mathbb{N}$.

\item En déduire la nature de la suite $\left(u_n\right)$.

\item Exprimer $u_n$ en fonction de $n$ , pour tout $n\in\mathbb{N}$.
\end{enumerate}

\item Pour tout $n\in\mathbb{N}$ (avec $n\neq 0$) , on désigne par $S_n$ le coût total en euros d'un puits de $n$ mètres.\\
Déterminer le coût total d'un puits de $n$ mètres.

\item Déterminer la profondeur maximale que l'on peut atteindre avec le crédit de \numprint{414000} euros.

\end{enumerate}

%\subsection{}%bac ES Amérique du Sud novembre 2003
%Monsieur X a placé 2 000 \euro~ le 31 décembre 2002 sur son livret bancaire, 
%à intérêts composés au taux annuel de $3,5\:\%$ (ce qui signifie que, chaque 
%année, les intérêts sont ajoutés au capital et produisent à leur tour des 
%intérêts). \`A partir de l'année suivante, il prévoit de placer, chaque 
%31 décembre, 700 \euro~ supplémentaires sur ce livret. \\
%On désigne par $C_n$ le capital, exprimé en euros, disponible le 
%$1\textsuperscript{er}$ janvier de l'année (2003 + $n$), o\`u $n$ est un entier naturel. 
%Ainsi, on a : 
%
%\[\text{C}_0 = \nombre{2000}.\] 
%
%\begin{enumerate} \item \begin{enumerate} \item Calculer le capital disponible le 
%$1\textsuperscript{er}$ janvier 2004. 
%
%\item \'Etablir, pour tout entier naturel $n$, une relation entre $C_{n + 1}$ 
%et $C_n$. 
%
%\end{enumerate} 
%
%\item Pour tout entier naturel $n$, on pose : 
%
%\[u_n = C_n + 20\:000.\] 
%
%\begin{enumerate} \item Démontrer que la suite $\left(u_n\right)$ est une suite 
%géométrique dont on déterminera la raison. 
%
%\item Exprimer $u_n$ en fonction de $n$. 
%
%\item En déduire que, pour tout entier naturel $n$, on a : 
%
%\[C_n = \nombre{22000} \times (1,035)^n - \nombre{20000}.\] 
%
%\item Calculer le capital disponible le $1\textsuperscript{er}$ janvier 2008 (on arrondira 
%le résultat à l'euro pr\`es). 
%
%\end{enumerate} 
%
%\item Le premier janvier 2008, Monsieur X retirera alors le capital disponible de la 
%banque pour financer un voyage dont le co\^ut (supposé fixe) est 
%de \nombre{6000}~ \euro. Il paiera cette somme en 4 mensualités qui seront 4 termes 
%consécutifs d'une suite arithmétique de raison 800 \euro. 
%
%Calculer le montant de chacune de ces 4 mensualités. 
%
%\end{enumerate} 



\subsection{}
On considère deux suites $\left(u_n\right)$ et $\left(v_n\right)$ définies pour tout $n\in\mathbb{N}$ par :
\[u_n=\dfrac{3\times 2^n-4n+3}{2}\text{ et }v_n=\dfrac{3\times 2^n+4n-3}{2}.\]

\begin{enumerate}
\item On définit la suite $\left(w_n\right)$ par $w_n=u_n+v_n$.\\

\begin{enumerate}
\item Montrer que la suite $\left(w_n\right)$ est géométrique. Préciser sa raison, son premier terme ainsi que son sens de variation.

\item Donner le terme général de $w_n$.
\end{enumerate}

\item  On définit la suite $\left(t_n\right)$ par $t_n=u_n-v_n$.\\

\begin{enumerate}
\item Montrer que la suite $\left(t_n\right)$ est arithmétique. Préciser sa raison, son premier terme ainsi que son sens de variation.

\item Donner le terme général de $t_n$.
\end{enumerate}

\item Donner le terme général de $u_n$ (indication : calculer $w_n+t_n$)

\item Donner le terme général de $v_n$.


\end{enumerate}

\subsection{}

\begin{enumerate}
\item Soient $u$ et $v$ deux réels.
\begin{enumerate}
\item Développer le produit $(x – u)(x – v)$.

\item En déduire que les réels $u$ et $v$ sont les racines du polynôme $x^2 – Sx + P$ où $S = u + v$ et $P = uv$.
\end{enumerate}

\item Existe-t-il deux nombres réels $u$ et $v$ :
\begin{enumerate}
\item dont le produit est 6 et la somme 4 ?

\item dont le produit est 6 et la somme 8 ?


\end{enumerate}

\item Écrire en langage naturel un algorithme qui permet de déterminer deux entiers dont la somme et le produit sont deux réels fixés et entrés par l’utilisateur.

\end{enumerate}

\subsection{\textcolor{blue}{Équation bicarrée}}
On veut résoudre l'équation  $(E) : ~x^4-9x^2+14=0$
\begin{enumerate}
\item On pose $X=x^2$.\\
Écrire l'équation $(E)$ en fonction de $X$.

\item Résoudre l'équation d'inconnue $X$.

\item En déduire les quatre solutions de $(E)$.
\end{enumerate}


\subsection{}
Soit $f$ la fonction définie sur $\mathscr{D}=\mathbb{R}\setminus\left\{-3\right\}$ par \\
$f(x)=\dfrac{x-1}{x+3}$ et soit $g$ la fonction définie sur $\mathbb{R}$ par \\
$g(x)=-x-5$.
\begin{enumerate}
\item Pourquoi -3 est-il une valeur interdite pour $f$ ?

\item On donne la courbe représentative de $f$.\\
Tracer sur ce graphique la courbe représentative de $g$.
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\item Résoudre l'inéquation $f(x)\geqslant g(x)$.\\
En déduire la position relative des deux courbes (c'est-à-dire trouver pour quelles valeurs de $x$ $\mathscr{C}_f$ est au-dessus de $\mathscr{C}_g$ et en dessous).
\end{enumerate}

\end{multicols}





\subsection{\textcolor{blue}{Tapis de Sierpinski}}
Dans un lointain futur, sur la planète Thêta, vivent Horion et Apolline au sein d’une colonie d’humains, gouvernée avec bienveillance par une I.A. Mais tout n’est pas si paisible sur la planète Thêta\dots
\og{}Le tapis est mité\fg{} constate Horion.\og{}Tu pourrais au moins
cacher ta joie\fg{} s’emporte Apolline qui adorait son tapis laineux.\\
Le tapis carré de 1 m de côté, ravissant, selon les goûts d’Apolline, hideux selon les critères d’Horion, est effectivement attaqué par des mites.
\og{}Il s’agit de mites de l’espèce Waclaw\fg{}, croit bon de préciser l’I.A. à qui personne n’avait rien demandé, exaspérant davantage Apolline.\\
Les mites ont leur stratégie ; le premier jour, elles s’attaquent à la partie centrale du tapis : si on partage le tapis
carré en 9 carrés de même aire, les mites mangent le carré central. Le second jour, elles s’attaquent de la même
manière aux 8 carrés autour du carré manquant, et ainsi de suite pour les autres jours.\\
(Remarque : la figure \og{}limite\fg{} qu'on obtient ainsi d'appelle le \og{}tapis de Sierpinski\fg{}, du nom du mathématicien polonais Waclaw Franciszek Sierpinski 1882-1961)



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\end{multicols}

\end{center}

Horion, ravi, commente : \og{}Au final, cela fera un joli tapis en dentelle, et dans quelques jours elles feront des troussi petits qu’on ne les verra plus. \fg{}.\\
Apolline lui répond, furieuse : \og{}Le tapis va disparaître complètement ! \fg{}. Horion n’en espérait pas tant !\\
Étudions ces deux affirmations ;\\
Pour tout entier $n\geqslant$, on désigne par $c_n$ la longueur, en m, du côté d’un trou carré qui apparaît au $n$-ième jour et $A_n$ l’aire, en m$^2$, de la surface totale mangée par les mites après $n$ jours.
\begin{enumerate}
\item Montrer que $\left(c_n\right)$ est une suite géométrique et, pour tout entier $n\geqslant$, exprimer $c_n$ en fonction de $n$.

\item Quelle est la valeur de $A_1$ ?\\
Vérifier ensuite que $A_2=\dfrac{17}{81}$

\item Chaque jour, les mites mangent $\dfrac{1}{9}$ de la partie du tapis restante.\\
Montrer que, pour tout entier $n\geqslant$, on a $A_{n+1}=\dfrac{8}{9}A_n+\dfrac{1}{9}$.

\item 
\begin{enumerate}
\item Pour tout entier $n\geqslant$, on pose $B_n= A_n-1$. Montrer que $\left(B_n\right)$ est une suite géométrique.

\item Pour tout entier $n\geqslant$, en déduire l’expression de $B_n$, puis de $A_n$ en fonction de $n$.
\end{enumerate}

\item 
\begin{enumerate}
\item À partir de combien de jours la surface du tapis sera-t-elle mangée à 99\:\% au moins ? (À trouver avec une calculatrice)\\
(On admet que $A_n$ tend vers 1 lorsque $n$ tend vers $+\infty$ et que la suite $\left(A_n\right)$ est croissante)

\item Sachant qu’à une distance supérieure à 1 m, un oeil humain normalement constitué ne peut distinguer de détail sur un tapis inférieur à 0,5 mm, au bout de combien de jours ne verra-t-on plus les nouveaux trous formés ? (À trouver à la calculatrice)\\
(On admet que $c_n$ tend vers 0 lorsque $n$ tend vers $+\infty$.)
\end{enumerate}

\item Que penser de chacune des affirmations de Horion et d’Apolline ?
\end{enumerate}

\label{fin}
\end{document}  