% récurrence et suites
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%%%     créée le Tue, 16 Sep 2008 10:35:26 +0200                 %%%
%%%     sur http://allken-bernard.org/pierre/phpmyexercices      %%%
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\lfoot{\small \textit{Lycée Joachim du Bellay}}
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%%%     Exercice 168
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Soit $(u_n)$ une suite arithmétique dont on notera $r$ la raison.
Sachant que $u_0=2$ et $r=-3$, calculer $u_{10}$ et $u_{20}$.
\end{exercice}


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%%%     Exercice 169
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Soit $(u_n)$ une suite arithmétique.
Sachant que $u_0=2$ et $u_1=5$, calculer $u_2$ et $u_5$.
\end{exercice}


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%%%     Exercice 170
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Soit $(u_n)$ une suite arithmétique.
Sachant que $u_5=17$ et $u_{10}=12$, calculer $u_0$ et $u_1$.
\end{exercice}


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%%%     Exercice 171
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Soit $(u_n)$ une suite géométrique de raison $r=\frac{1}{4}$.
Sachant que $u_0=32$, calculer $u_1$, $u_2$, $u_3$.
\end{exercice}


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%%%     Exercice 173
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Soit $(u_n)$ une suite géométrique. Sachant que $u_0=-1$ et $u_{10}=1$, calculer $u_0$.
\end{exercice}


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%%%     Exercice 172
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Soit $(u_n)$ une suite géométrique. Sachant que $u_0=3$ et $u_2=12$, calculer $u_1$ et $u_5$.
\end{exercice}


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%%%     Exercice 141
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Poursuivre en ayant le moins d'imagination possible~:
\[1,3,6,10,15,21,\ldots\]

\end{exercice}


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%%%     Exercice 142
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Poursuivre en ayant le moins d'imagination possible~:
\[1,1,2,3,5,8,13,21,\ldots\]

\end{exercice}


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%%%     Exercice 143
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Poursuivre en ayant le moins d'imagination possible~:
\[1,11,21,1211,111221,312211,\ldots\]

\end{exercice}


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%%%     Exercice 150
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Pour tout $n\in\mathbf N^*$, on pose~:
\[u_n=1+3+5+\cdots+(2n-1)\]
\begin{enumerate}
\item Calculer $u_n$ pour $n=1,2,3,4,5$.
\item Conjecturer une formule simple pour $u_n$.
\item Démontrer que cette formule est vraie.
\end{enumerate}
\end{exercice}


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%%%     Exercice 151
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Soit $(u_n)_{n\in\mathbf N}$ la suite définie par $u_0=0$ et :
\[\forall n\in\mathbf N,\; u_{n+1}=2u_n+1\]
\begin{enumerate}
\item Calculer $u_n$ pour $n=0,1,2,3,4$.
\item Conjecturer une formule simple pour $u_n$.
\item Démontrer que cette formule est vraie.
\end{enumerate}
\end{exercice}


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%%%     Exercice 153
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Posons $u_n=1+2+4+\cdots+2^n$ pour tout $n\in\mathbf N$.
\begin{enumerate}
\item Calculer $u_n$ pour $n=0,1,2,3,4,5$.
\item Conjecturer une formule simple pour $u_n$.
\item Démontrer que cette formule est vraie.
\end{enumerate}
\end{exercice}


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%%%     Exercice 154
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Pour tout entier $n\ge 3$, on note $d_n$ le nombre de diagonales d'un polygone à $n$ sommets.
\begin{enumerate}
\item Calculer $d_n$ pour $n=3,4,5,6$.
\item Démontrer que, pour tout entier $n\ge 3$~:
\[d_n=\frac{n(n-3)}{2}\]
\item Combien de diagonales possède un myriagone (10000 sommets)~?
\end{enumerate}
\end{exercice}


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%%%     Exercice 155
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Soit $(u_n)_{n\in\mathbf N}$ la suite définie par :
\[\left\{\begin{array}{l} u_0=1\\ \forall n\in\mathbf N,\;u_{n+1}=\frac{1}{3}(u_n+4n+6)\end{array}\right.\]
\begin{enumerate}
\item Calculer $u_n$ pour $n=0,1,2,3$.
\item Démontrer que :
\[\forall n\in\mathbf N,\;u_n=2n+\frac{1}{3^n}\]
\end{enumerate}
\end{exercice}


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%%%     Exercice 156
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]On place 1000 euros, avec intérêts annuels composés (c'est-à-dire qu'à la fin de chaque année, les intérêts sont incorporés au capital), à un taux de $2\%$.
On note $u_n$ la somme, en euros, dont on dispose à la fin de la $n$-ème année, en convenant que $u_0=1000$.
\begin{enumerate}
\item Calculer $u_n$ pour $n=0,1,2$.
\item Calculer $u_{n+1}$ en fonction de $u_n$. Quelle est la nature de la suite $(u_n)$~?
\item En déduire une expression de $u_n$ en fonction de $n$.
\item Aprés combien d'années dispose-t-on d'au moins 2000 euros~?
\end{enumerate}

\end{exercice}


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%%%     Exercice 163
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]On place 2000 euros, avec intérêts annuels composés (c'est-à-dire qu'à la fin de chaque année, les intérêts sont incorporés au capital), à un taux de $4\%$, et on ajoute au capital 1000 euros au début de chaque année suivante.
On note $u_n$ la somme, en euros, dont on dispose à la fin de la $n$-ème année, en convenant que $u_0=2000$.
\begin{enumerate}
\item Calculer $u_n$ pour $n=0,1,2$.
\item Calculer $u_{n+1}$ en fonction de $u_n$. Quelle est la nature de la suite $(u_n)$~?
\item En déduire une expression de $u_n$ en fonction de $n$.
\item Aprés combien d'années dispose-t-on d'au moins 29000 euros~?
\end{enumerate}

\end{exercice}


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%%%     Exercice 152
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Soit $(u_n)_{n\in\mathbf N}$ la suite définie par :
\[\left\{\begin{array}{l} u_0=4\\ \forall n\in\mathbf N,\;u_{n+1}=u_n^2-2\end{array}\right.\]
Démontrer que :
\[\forall n\in\mathbf N,\;u_n=(2+\sqrt 3)^{2^n}+(2-\sqrt 3)^{2^n}\]
\end{exercice}


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%%%     Exercice 157
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Soit $(u_n)$ une une suite qui est à la fois arithmétique de raison $r$ et géométrique de raison $q$. Démontrer que $(u_n)$ est constante.
\end{exercice}


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%%%     Exercice 158
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Exprimer $a_n$ en fonction de $n$.
\[\left\{\begin{array}{l}a_0=0\\\forall n\in\mathbf N,\;a_{n+1}=-a_n+1\end{array}\right.\]
\end{exercice}


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%%%     Exercice 159
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Exprimer $b_n$ en fonction de $n$.
\[\left\{\begin{array}{l}b_0=1\\\forall n\in\mathbf N,\;b_{n+1}=5b_n+6\end{array}\right.\]
\end{exercice}


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%%%     Exercice 160
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Exprimer $c_n$ en fonction de $n$.
\[\left\{\begin{array}{l}c_0=-2\\\forall n\in\mathbf N,\;c_{n+1}=2c_n+2\end{array}\right.\]
\end{exercice}


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%%%     Exercice 161
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Exprimer $d_n$ en fonction de $n$.
\[\left\{\begin{array}{l}d_0=0\\\forall n\in\mathbf N,\;d_{n+1}=-4d_n-1\end{array}\right.\]
\end{exercice}


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%%%     Exercice 162
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Exprimer $e_n$ en fonction de $n$.
\[\left\{\begin{array}{l}e_0=-1\\\forall n\in\mathbf N,\;e_{n+1}=2e_n+4\end{array}\right.\]
\end{exercice}


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%%%     Exercice 164
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Exprimer $u_n$ en fonction de $n$.
\[\left\{\begin{array}{l}u_0=0\\u_1=1\\\forall n\in\mathbf N,\;u_{n+2}=3u_{n+1}-2u_n\end{array}\right.\]
\end{exercice}


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%%%     Exercice 165
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Exprimer $v_n$ en fonction de $n$.
\[\left\{\begin{array}{l}v_0=1\\ v_1=1\\\forall n\in\mathbf N,\;v_{n+2}=v_{n+1}+v_n\end{array}\right.\]
\end{exercice}


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%%%     Exercice 166
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Exprimer $w_n$ en fonction de $n$.
\[\left\{\begin{array}{l}w_0=0\\ w_1=1\\\forall n\in\mathbf N,\;w_{n+2}=6w_{n+1}-9w_n\end{array}\right.\]
\end{exercice}


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%%%     Exercice 167
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]On définit une suite par :
\[\left\{\begin{array}{l}u_0=2\\
\forall n\in\mathbf N,\;u_{n+1}=-2u_n+n^2-2\end{array}\right.\]
Montrer qu'il existe des réels $a,b,c$ tels que la suite définie par :
\[v_n=u_n+an^2+bn+c\]
soit géométrique. En déduire une expression simple de $u_n$ en fonction $n$.
\end{exercice}





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%%%     Exercice 174
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Soit $(u_n)$ la suite définie par $u_0=0$ et $u_{n+1}=-\frac{2}{3}u_n+1$. Calculer $u_n$ en fonction de $n$.
\end{exercice}


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%%%     Exercice 175
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Soit $(u_n)$ la suite définie par $u_0=5$ et $u_{n+1}=2u_n+3$. Calculer $u_n$ en fonction de $n$.
\end{exercice}


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%%%     Exercice 176
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Soit $(u_n)$ la suite définie par $u_0=1$ et $u_{n+1}=\frac{1}{2}u_n+\frac{1}{4}$. Calculer $u_n$ en fonction de $n$.
\end{exercice}


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%%%     Exercice 177
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{exercice}
\nopagebreak[5]Soit $(u_n)$ la suite définie par $u_0=6$ et $u_{n+1}=\frac{1}{3}u_n+2$. Calculer $u_n$ en fonction de $n$.
\end{exercice}


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%%%     Exercice 178
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Soit $(u_n)$ la suite définie par $u_0=0$ et $u_{n+1}=\frac{2u_n+3}{u_n+4}$. On pose $v_n=\frac{u_n-1}{u_n+3}$. Montrer que $(v_n)$ est géométrique. En déduire $u_n$ en fonction de $n$.
\end{exercice}


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%%%     Exercice 179
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{exercice}
\nopagebreak[5]Soit $(u_n)$ la suite définie par $u_0=1$ et $u_{n+1}=\sqrt{2+u_n^2}$. On pose $v_n=u_n^2$. Montrer que $(v_n)$ est géométrique. En déduire $u_n$ en fonction de $n$.
\end{exercice}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%     Exercice 180
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{exercice}
\nopagebreak[5]Soit $(u_n)$ la suite définie par $u_1=\frac{1}{3}$ et $u_{n+1}=\frac{n+1}{3n}u_n$. On pose $v_n=\frac{u_n}{n}$ pour tout $n\in\mathbf N^*$. Montrer que $(v_n)$ est géométrique. En déduire $u_n$ en fonction de $n$.
\end{exercice}


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%%%     Exercice 181
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{exercice}
\nopagebreak[5]Soit $(u_n)$ la suite définie par $u_0=1$, $u_1=3$, et $u_{n+2}=10u_{n+1}-9u_n$. Calculer $u_n$ en fonction de $n$.
\end{exercice}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%     Exercice 182
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Soit $(u_n)$ la suite définie par $u_0=0$, $u_1=1$, et $u_{n+2}=7u_{n+1}+8u_n$. Calculer $u_n$ en fonction de $n$.
\end{exercice}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%     Exercice 144
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{exercice}
\nopagebreak[5]Démontrer que $\forall n\in\mathbf N,\; 2^n> n$.
\end{exercice}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%     Exercice 145
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{exercice}
\nopagebreak[5]Démontrer que $\forall n\in\mathbf N,\; 3^n> n$.
\end{exercice}


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%%%     Exercice 146
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Notons $u_n$ le nombre de poignées de main lorsque $n$ personnes ($n\ge 2$) se serrent la main.
\begin{enumerate}
\item Calculer $u_n$ pour $n=2,3,4,5$.
\item Démontrer par récurrence que~:
\[u_n=\frac{n(n-1)}{2}\]
\end{enumerate}
\end{exercice}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%     Exercice 147
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Démontrer que pour tout entier $n\ge 1$~:
\[1+2+3+\cdots+n=\frac{n(n+1)}{2}\]

\end{exercice}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%     Exercice 148
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Démontrer que pour tout entier $n\ge 1$~:
\[1^2+2^2+3^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}\]

\end{exercice}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%     Exercice 149
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{exercice}
\nopagebreak[5]Démontrer que pour tout entier $n\ge 1$~:
\[1^3+2^3+3^3+\cdots+n^3=\left(\frac{n(n+1)}{2}\right)^2\]

\end{exercice}

\label{fin}
\end{document}