% sommes
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%%%     créée le Thu, 25 Sep 2008 00:20:10 +0200                 %%%
%%%     sur http://allken-bernard.org/pierre/phpmyexercices      %%%
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\lfoot{\small \textit{Lycée Joachim du Bellay}}
\cfoot{\small \textit{Mathématiques, prépa ECE1}}
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%%%     Exercice 225
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]\'Ecrire le nombre suivant sans le symbole $\Sigma$ (on ne demande pas de le calculer)~:
\[\sum_{i=1}^{n} \ln(i)\]
\end{exercice}


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%%%     Exercice 226
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]\'Ecrire le nombre suivant sans le symbole $\Sigma$ (on ne demande pas de le calculer)~:
\[\sum_{i=0}^{n-1} \ln(i+1)\]
\end{exercice}


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%%%     Exercice 227
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]\'Ecrire le nombre suivant sans le symbole $\Sigma$ (on ne demande pas de le calculer)~:
\[\sum_{i=2}^{n+1} \ln(i-1)\]
\end{exercice}


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%%%     Exercice 228
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]\'Ecrire le nombre suivant sans le symbole $\Sigma$ puis le calculer~:
\[\sum_{i=1}^{n} 1\]
\end{exercice}


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%%%     Exercice 229
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]\'Ecrire le nombre suivant sans le symbole $\Sigma$ puis le calculer~:
\[\sum_{i=1}^{2008} (-1)^i\]
\end{exercice}


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%%%     Exercice 230
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]\'Ecrire la somme suivante à l'aide du symbole $\Sigma$ (on ne demande pas de la calculer)~:
\[\ln(2)+\ln(3)+\ln(4)+\cdots+\ln(42)\]

\end{exercice}


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%%%     Exercice 231
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]\'Ecrire la somme suivante à l'aide du symbole $\Sigma$ (on ne demande pas de la calculer)~:
\[1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\cdots+\frac{1}{n}\]

\end{exercice}


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%%%     Exercice 232
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]\'Ecrire la somme suivante à l'aide du symbole $\Sigma$ (on ne demande pas de la calculer)~:
\[\frac{1}{1+1!}+\frac{4}{1+2!}+\frac{9}{1+3!}+\cdots+\frac{100^2}{1+100!}\]

\end{exercice}


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%%%     Exercice 233
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]\'Ecrire la somme suivante à l'aide du symbole $\Sigma$ (on ne demande pas de la calculer)~:
\[1-2+3-4+\cdots+103\]

\end{exercice}


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%%%     Exercice 234
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]\'Ecrire la somme suivante à l'aide du symbole $\Sigma$ (on ne demande pas de la calculer)~:
\[-1+2-3+4+\cdots-103\]

\end{exercice}


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%%%     Exercice 235
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]\'Ecrire la somme suivante à l'aide du symbole $\Sigma$ (on ne demande pas de la calculer)~:
\[2+4+6+\cdots + 248\]

\end{exercice}


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%%%     Exercice 236
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]\'Ecrire la somme suivante à l'aide du symbole $\Sigma$ (on ne demande pas de la calculer)~:
\[1+3+5+\cdots+249\]

\end{exercice}


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%%%     Exercice 237
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]\'Ecrire la somme suivante à l'aide du symbole $\Sigma$ (on ne demande pas de la calculer)~:
\[1000+1010+1020+\cdots+1540\]

\end{exercice}


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%%%     Exercice 238
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]\'Ecrire la somme suivante à l'aide du symbole $\Sigma$ (on ne demande pas de la calculer)~:
\[1+2+4+8+16+\cdots+1024\]

\end{exercice}


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%%%     Exercice 239
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]\'Ecrire la somme suivante à l'aide du symbole $\Sigma$ (on ne demande pas de la calculer)~:
\[3-6+9-12+\cdots+303\]

\end{exercice}


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%%%     Exercice 240
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]\'Ecrire la somme suivante en faisant en sorte que la première valeur de l'indice soit~$0$~:
\[\sum_{i=10}^{20} i\]

\end{exercice}


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%%%     Exercice 241
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]\'Ecrire la somme suivante en faisant en sorte que la première valeur de l'indice soit~$0$~:
\[\sum_{k=-4}^{180} \frac{k}{k+5}\]

\end{exercice}


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%%%     Exercice 242
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]\'Ecrire la somme suivante en faisant en sorte que la première valeur de l'indice soit~$0$~:
\[\sum_{i=2}^{45} 1\]

\end{exercice}


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%%%     Exercice 243
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]\'Ecrire la somme suivante en faisant en sorte que la première valeur de l'indice soit~$0$~:
\[\sum_{i=1}^{n} i\]
Même question en remplaçant $0$ par $49$.
\end{exercice}


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%%%     Exercice 244
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Changer d'indice dans la somme suivante pour que le terme général soit plus simple~:
\[\sum_{i=0}^{n} \frac{(i+1)^2+3}{1+\sqrt{i+1}}\]
\end{exercice}


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%%%     Exercice 245
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Changer d'indice dans la somme suivante pour que le terme général soit plus simple~:
\[\sum_{k=3}^{n+2} \frac{x^{k-3}}{(k-3)!}\]
\end{exercice}


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%%%     Exercice 246
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Changer d'indice dans la somme suivante pour que le terme général soit plus simple~:
\[\sum_{i=1}^{n+1} (i-1)2^i+3^{i-1}\]
\end{exercice}


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%%%     Exercice 247
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Calculer~:
\[\sum_{i=1}^{n} (2i^3+3i^2+i-1)\]

\end{exercice}


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%%%     Exercice 248
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Calculer~:
\[\sum_{i=0}^{n} \left(2^i+\left(\frac{1}{2}\right)^i\right)\]

\end{exercice}


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%%%     Exercice 249
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Calculer~:
\[\sum_{i=0}^{n} \frac{1}{3^i}\]

\end{exercice}


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%%%     Exercice 250
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Calculer~:
\[\sum_{i=0}^{n} 2^{2i+1}\]

\end{exercice}


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%%%     Exercice 251
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Calculer~:
\[\sum_{i=n}^{2n} i\]

\end{exercice}


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%%%     Exercice 252
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Calculer~:
\[\sum_{i=n}^{2n} 1\]

\end{exercice}


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%%%     Exercice 253
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Calculer~:
\[\sum_{i=n}^{2n} (i+1+n)\]

\end{exercice}


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%%%     Exercice 254
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Calculer~:
\[\sum_{i=1}^{n} ((-2)^i+i)\]

\end{exercice}


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%%%     Exercice 255
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Calculer~:
\[1+2+3+\cdots+100\]
\end{exercice}


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%%%     Exercice 256
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Calculer~:
\[2+4+6+\cdots+100\]
\end{exercice}


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%%%     Exercice 257
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Calculer~:
\[1+3+5+\cdots+99\]
\end{exercice}


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%%%     Exercice 258
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Calculer~:
\[\sum_{i=-n}^{n} (i+1)\]

\end{exercice}


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%%%     Exercice 259
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Calculer~:
\[\sum_{i=1}^n 4i(i^2-1)\]

\end{exercice}


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%%%     Exercice 260
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]En partant de la somme géométrique :
\[\forall x\in\mathbf R\setminus\{1\},\quad \sum_{i=0}^n x^i=\frac{1-x^{n+1}}{1-x}\]
retrouver la somme géométrique dérivée :
\[\forall x\in\mathbf R\setminus\{1\},\quad \sum_{i=0}^n ix^{i-1}=\ldots\]
En déduire $\displaystyle{\sum_{i=0}^n ix^i}$.
\end{exercice}


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%%%     Exercice 261
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Soit $x\in\mathbf R\setminus\{1\}$.
\begin{enumerate}
\item Calculer la \og{}somme géométrique dérivée deux fois\fg{}~:
\[\sum_{i=0}^n i(i-1) x^{i-2}\]
\item En déduire $\displaystyle{\sum_{i=0}^n i(i-1) x^i}$
\item En déduire $\displaystyle{\sum_{i=0}^n i^2 x^2}$
\end{enumerate}
\end{exercice}


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%%%     Exercice 262
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Calculer~:
\[1-x+x^2-x^3+\cdots+(-1)^n x^n\]
\end{exercice}


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%%%     Exercice 263
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{exercice}
\nopagebreak[5]Calculer~:
\[1+a^2+a^4+\cdots + a^{2n}\]
\end{exercice}


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%%%     Exercice 264
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Calculer~:
\[\sum_{k=1}^{2n} k(2k-1)(k+1)\]
\end{exercice}


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%%%     Exercice 265
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Calculer~:
\[-8\cdot 7^2+8\cdot 7^3-8\cdot 7^4+\cdots -8\cdot 7^{1000}\]
\end{exercice}


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%%%     Exercice 266
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Calculer~:
\[\sum_{i=0}^n \frac{3}{10^i}\]
\end{exercice}


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%%%     Exercice 267
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{exercice}
\nopagebreak[5]Calculer~:
\[\sum_{i=0}^{2n} 3\cdot 4^{i+1}\]
\end{exercice}


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%%%     Exercice 268
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{exercice}
\nopagebreak[5]Calculer~:
\[\sum_{i=0}^{n}\frac{5\cdot 2^i}{3^{i+1}}\]
\end{exercice}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%     Exercice 269
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{exercice}
\nopagebreak[5]Calculer~:
\[\sum_{i=0}^{n+1} 3^{2i+1}\]
\end{exercice}


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%%%     Exercice 270
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Calculer~:
\[\sum_{i=3}^{n+1} \frac{2^i}{3^{i+2}}\]
\end{exercice}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%     Exercice 271
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{exercice}
\nopagebreak[5]Calculer~:
\[\sum_{i=1}^n (5\cdot 2^i+2\cdot 3^i)\]
\end{exercice}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%     Exercice 272
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{exercice}
\nopagebreak[5]Calculer~:
\[\sum_{k=3}^{2n} 2^{3k+1}\cdot \frac{3^{k+1}}{4^k}\]
\end{exercice}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%     Exercice 273
%%%     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 $x\in \mathbf R\setminus \{0,-1\}$~:
\[\frac{1}{x(x+1)}=\frac{1}{x}-\frac{1}{x+1}\]
En déduire~:
\[\sum_{i=1}^n \frac{1}{i(i+1)}\]
\end{exercice}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%     Exercice 274
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Soit $(a_n)_{n\in\mathbf N}$ une suite. Montrer par récurrence que~:
\[\forall n\in\mathbf N^*,\;\sum_{k=0}^{n-1} k(a_k-a_{k+1})=\left(\sum_{k=1}^n a_k\right)-n a_n\]
\end{exercice}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%     Exercice 275
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Calculer~:
\[\sum_{i=1}^{50} \frac{1}{i}-\sum_{k=3}^{51} \frac{1}{k}\]
\end{exercice}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%     Exercice 276
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{exercice}
\nopagebreak[5]Calculer~:
\[\sum_{i=-2}^{40} \left(10i+4\right)\]
\end{exercice}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%     Exercice 277
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Calculer~:
\[\sum_{i=1}^{n} \ln\left(\frac{i}{i+1}\right)\]
\end{exercice}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%     Exercice 278
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{exercice}
\nopagebreak[5]Soit $(a_n)_{n\in\mathbf N}$ la suite définie par :
\[\left\{\begin{array}{l}
a_0=-3\\
\forall n\in\mathbf N, a_{n+1}=5a_n-2
\end{array}\right.\]
Calculer~:
\[\sum_{i=0}^{n} a_i\]
\end{exercice}


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%%%     Exercice 279
%%%     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 :
\[\left\{\begin{array}{l}
u_0=0\\
u_1=1\\
\forall n\in\mathbf N, u_{n+2}=3u_{n+1}-2u_n
\end{array}\right.\]
Calculer~:
\[\sum_{i=0}^{n-1} u_i\]
\end{exercice}


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%%%     Exercice 280
%%%     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 :
\[\left\{\begin{array}{l}
u_0=0\\
u_1=1\\
\forall n\in\mathbf N, u_{n+2}=10u_{n+1}-25u_n
\end{array}\right.\]
Calculer~:
\[\sum_{i=0}^{n-1} u_i\]
\end{exercice}

\label{fin}
\end{document}