feat(1G_spec): exercices sur les suites
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1G_EnsSci/03_suites_arithmetiques/exercises.tex
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1G_EnsSci/03_suites_arithmetiques/exercises.tex
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\begin{exercise}[subtitle={Carrés de Pierre}, step={1}, origin={}, topics={ Modélisation Discrète }, tags={ modélisation, suite}, mode={\searchMode}]
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\begin{minipage}{ 0.45\textwidth }
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Pierre joue avec des mosaïques de couleur. Il dispose ses mosaïques pour obtenir des « carrés »
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Il voudrait savoir à l’avance combien de mosaïques il lui faut pour fabriquer n’importe quel « carré ». Comment l’aider ?
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\end{minipage}
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\hfill
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\begin{minipage}{ 0.5\textwidth }
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\includegraphics[scale=0.8]{./fig/carre.pdf}
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\end{minipage}
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\end{exercise}
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\begin{exercise}[subtitle={Abonnement téléphone}, step={1}, origin={}, topics={ Modélisation Discrète }, tags={ modélisation, suite }, mode={\trainMode}]
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Abdou se rend chez un opérateur téléphonique pour choisir un abonnement. On lui propose 3 forfaits:
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\begin{itemize}
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\item Forfait \textbf{Liberté}: \dotfill 36\euro par mois, textos illimités.
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\item Forfait \textbf{Zen}: \dotfill 20\euro par mois, 5 centimes par textos.
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\item Forfait \textbf{Proche}: \dotfill 20 centimes par textos
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\end{itemize}
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\begin{enumerate}
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\item Abdou estime qu'il envoie 150 textos par mois. Quel forfait doit-il choisir?
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\item Une copine, Roubouanti, envoie en moyenne 300 textos par mois et a choisi le forfait \textbf{Liberté}. A-t-elle fait le bon choix?
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\item Expliquer en une phrase, pour chaque forfait, comment calculer le prix à payer pour n'importe quel nombre de texto envoyé.
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\item Abdou souhaite en savoir plus. Il se demande à partir de combien de textos envoyé, il est plus intéréssant de choisir le forfait \textbf{Liberté}. Aidez le à répondre à sa question.
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\end{enumerate}
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\end{exercise}
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\begin{exercise}[subtitle={Modélisation par une suite arithmétique}, step={2}, origin={livre scolaire}, topics={ Modélisation Discrète }, tags={ modélisation, suite }, mode={\trainMode}]
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\begin{enumerate}
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\item À la salle de sport, un adhérent charge sa barre de squat, qui pèse 20 kg, avec 40 kg de poids de chaque côté pour sa première série. Pour les séries suivantes, il baisse la charge de son équipement qui pèse alors 90 kg, puis 80 kg, puis 70 kg, etc.
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Peut-on modéliser cette situation par une suite arithmétique ?
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\item La réserve de noisettes d'un écureuil diminue chaque jour de l'hiver. Elle débute à 156 noisettes, puis atteint 122 noisettes au bout d'un mois, 86 au bout de 2 mois.
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Sur la durée de l'hiver, peut-on modéliser cette situation par une suite arithmétique ? Justifier.
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\end{enumerate}
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\end{exercise}
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\begin{exercise}[subtitle={Reconnaitre une suite arithmétique}, step={2}, origin={livre scolaire}, topics={ Modélisation Discrète }, tags={ modélisation, suite }, mode={\trainMode}]
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Les situations suivantes sont à croissance linéaire.
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\begin{enumerate}
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\item Dans un élevage de chèvres, la population au cours de quatre mois consécutifs s'élève à 76, puis 70, puis 64, puis 58 chèvres. Combien y a-t-il de chèvres le mois suivant ?
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\item La quantité d'eau dans un pluviomètre lors d'un orage est relevée toutes les dix minutes. Les valeurs relevées sont 15 mm ; 19 mm ; 23 mm ; 27 mm. Quelle valeur est relevée dix minutes plus tard ?
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\end{enumerate}
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\end{exercise}
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\begin{exercise}[subtitle={Transition démographique}, step={2}, origin={}, topics={ Modélisation Discrète }, tags={ modélisation, tableur, programmation }, mode={\searchMode}]
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\begin{multicols}{2}
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\begin{doc}{ Population française entre 1950 et 1980 selon l'INSEE }
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% \begin{tabular}{|c|c|}
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% \hline
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% Année & Population \\
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% \hline
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% 1950 & 41 647 258 \\
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% 1952 & 42 300 981 \\
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% 1954 & 42 885 138 \\
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% 1956 & 43 627 467 \\
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% 1958 & 44 563 043 \\
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% 1960 & 45 464 797 \\
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% 1962 & 46 422 000 \\
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% 1964 & 48 059 022 \\
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% 1966 & 48 953 792 \\
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% 1968 & 49 723 072 \\
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% 1970 & 50 528 219 \\
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% 1972 & 51 485 953 \\
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% 1974 & 52 320 725 \\
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% 1976 & 52 798 338 \\
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% 1978 & 53 271 566 \\
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% 1980 & 53 731 387 \\
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% \hline
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% \end{tabular}
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\begin{tabular}{cc|cc}
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\hline
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Année & Population & Année & Population \\
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\hline
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1950 & 41 647 258 & 1966 & 48 953 792 \\
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1952 & 42 300 981 & 1968 & 49 723 072 \\
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1954 & 42 885 138 & 1970 & 50 528 219 \\
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1956 & 43 627 467 & 1972 & 51 485 953 \\
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1958 & 44 563 043 & 1974 & 52 320 725 \\
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1960 & 45 464 797 & 1976 & 52 798 338 \\
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1962 & 46 422 000 & 1978 & 53 271 566 \\
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1964 & 48 059 022 & 1980 & 53 731 387 \\
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\hline
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\end{tabular}
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\end{doc}
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\begin{doc}{ INSEE }
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L’Institut national de la statistique et des études économiques dépend du ministère de l’Économie et des Finances. Son rôle est de collecter des données et de réaliser des analyses statistiques sur la société française et son évolution. L’Insee est chargé du recensement de la population française. Selon ses estimations, la population française est de \np{67 063 700} habitants au 1er janvier 2020.
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\end{doc}
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\begin{enumerate}
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\item Représenter graphiquement la population française en fonction du temps.
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\item Proposer un modèle pour faire une estimation de la population en 1990 puis en 2020.
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\item Comparer vos résultats à l'estimation de l'INSEE.
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\end{enumerate}
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\end{multicols}
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\end{exercise}
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\begin{exercise}[subtitle={Formule explicite}, step={3}, origin={}, topics={ Modélisation Discrète }, tags={ modélisation, tableur, programmation }, mode={\trainMode}]
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Calculer les valeurs de $u(0)$, $u(1)$, $u(2)$ et $u(10)$ pour les suites suivantes puis reconnaitre les paramètres et le sens de variation de ces suites arithmétiques
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\begin{multicols}{3}
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\begin{enumerate}
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\item $u(n) = 2 + 3n$
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\item $u(n) = 5n - 10$
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\item $u(n) = \dfrac{n - 4}{2}$
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\end{enumerate}
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\end{multicols}
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\end{exercise}
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\begin{exercise}[subtitle={Formule de récurrence}, step={3}, origin={}, topics={ Modélisation Discrète }, tags={ modélisation, tableur, programmation }, mode={\trainMode}]
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Calculer les valeurs de $u(0)$, $u(1)$, $u(2)$ et $u(10)$ pour les suites suivantes puis reconnaitre les paramètres et le sens de variation de ces suites arithmétiques
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\begin{multicols}{2}
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\begin{enumerate}
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\item $u(0) = 4$ et $u(n+1) = u(n) + 5$
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\item $u(0) = -7$ et $u(n+1) = u(n) + 10$
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\item $u(0) = 1$ et $u(n+1) = u(n) - 3$
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\end{enumerate}
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\end{multicols}
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\end{exercise}
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\begin{exercise}[subtitle={Placement à taux fixe}, step={4}, origin={}, topics={ Modélisation Discrète }, tags={ modélisation, tableur, programmation }, mode={\trainMode}]
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Guillaume décide de faire un placement à intérêts simples afin de prévoir l'achat d'une moto à 13000 €. Il place 9500 € en janvier 2022. À chaque début de mois, son capital est augmenté de 1,1\% du montant initial.
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On décide de modéliser le montant du placement par une suite noté $p(n)$ au bout de n mois après le 1er janvier 2022.
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\begin{enumerate}
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\item Déterminer les valeurs de $p(0)$, $p(1)$ et $p(2)$.
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\item Justifier que pour tout $n$ on a $p(n) = 104.5n+9500$
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\item Au bout de combien de mois, pourra-t-il acheter sa moto?
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\end{enumerate}
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\end{exercise}
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\begin{exercise}[subtitle={Remboursement}, step={4}, origin={}, topics={ Modélisation Discrète }, tags={ modélisation, tableur, programmation }, mode={\trainMode}]
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Maggie emprunte 50 € à son meilleur ami Axel. Pour le rembourser, elle décide de lui donner 10 € le lundi suivant, puis, chaque lundi, de lui donner 2 €.
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\begin{enumerate}
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\item Justifier que la situation peut être modélisée par une suite arithmétique dont on précisera le sens de $n$, le premier terme et la raison.
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\item Quelle somme Maggie aura-t-elle remboursée au bout de quatre semaines après son versement de 10 € ?
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\item Au bout de combien de temps Maggie aura-t-elle remboursé Axel intégralement ?
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\end{enumerate}
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\end{exercise}
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1G_EnsSci/03_suites_arithmetiques/fig/carre.pdf
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1G_EnsSci/03_suites_arithmetiques/fig/carre.pdf
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1G_EnsSci/03_suites_arithmetiques/fig/carre.svg
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1G_EnsSci/03_suites_arithmetiques/fig/carre.svg
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43
1G_EnsSci/03_suites_arithmetiques/index.rst
Normal file
43
1G_EnsSci/03_suites_arithmetiques/index.rst
Normal file
@@ -0,0 +1,43 @@
|
||||
suites arithmétiques
|
||||
####################
|
||||
|
||||
:date: 2025-09-30
|
||||
:modified: 2025-09-30
|
||||
:authors: Benjamin Bertrand
|
||||
:tags: représentation graphique, suite
|
||||
:category: 1G_EnsSci
|
||||
:summary: Modélisation de phénomènes linéaires avec les suites arithmétiques
|
||||
|
||||
|
||||
Éléments du programme
|
||||
=====================
|
||||
|
||||
Contenus
|
||||
--------
|
||||
|
||||
- Définition par la relation de récurrence.
|
||||
- Explicitation du terme de rang n.
|
||||
- Sens de variation.
|
||||
- Représentation graphique.
|
||||
|
||||
Capacités attendues
|
||||
-------------------
|
||||
|
||||
- Reconnaître un phénomène discret croissance linéaire et savoir le modéliser.
|
||||
- Calculer un terme de rang donné d'une suite arithmétique définie par une relation fonctionnelle ou une relation de récurrence.
|
||||
- Réaliser et exploiter la représentation graphique des termes d'une suite arithmétique
|
||||
- Résoudre un problème de seuil dans le cas d'une croissance linéaire.
|
||||
|
||||
Progression
|
||||
===========
|
||||
|
||||
Étape 1: Modélisation de phénomènes de croissances linéaires
|
||||
------------------------------------------------------------
|
||||
|
||||
Étape 2: Reconnaître une suite arithmétique
|
||||
-------------------------------------------
|
||||
|
||||
Étape 3: Utiliser une suite arithmétique
|
||||
----------------------------------------
|
||||
|
||||
|
||||
BIN
1G_EnsSci/03_suites_arithmetiques/plan_de_travail.pdf
Normal file
BIN
1G_EnsSci/03_suites_arithmetiques/plan_de_travail.pdf
Normal file
Binary file not shown.
53
1G_EnsSci/03_suites_arithmetiques/plan_de_travail.tex
Normal file
53
1G_EnsSci/03_suites_arithmetiques/plan_de_travail.tex
Normal file
@@ -0,0 +1,53 @@
|
||||
\documentclass[a4paper,12pt]{article}
|
||||
\usepackage{myXsim}
|
||||
|
||||
\author{Benjamin Bertrand}
|
||||
\title{suites arithmétiques - Plan de travail}
|
||||
\tribe{1G spécifique}
|
||||
\date{octobre 2025}
|
||||
|
||||
\pagestyle{empty}
|
||||
|
||||
\DeclareExerciseCollection{banque}
|
||||
\xsimsetup{
|
||||
}
|
||||
|
||||
|
||||
\begin{document}
|
||||
\maketitle
|
||||
|
||||
% Résumé
|
||||
|
||||
\bigskip
|
||||
|
||||
Savoir-faire de la séquence
|
||||
\begin{itemize}
|
||||
\item - Reconnaître un phénomène discret croissance linéaire et savoir le modéliser.
|
||||
\item - Calculer un terme de rang donné d'une suite arithmétique définie par une relation fonctionnelle ou une relation de récurrence.
|
||||
\item - Réaliser et exploiter la représentation graphique des termes d'une suite arithmétique
|
||||
\item - Résoudre un problème de seuil dans le cas d'une croissance linéaire.
|
||||
\end{itemize}
|
||||
|
||||
|
||||
\section{Modélisation par une suite arithmétique}
|
||||
|
||||
\listsectionexercises
|
||||
|
||||
\section{Reconnaître une suite arithmétique}
|
||||
|
||||
\listsectionexercises
|
||||
|
||||
\section{Utiliser une suite arithmétique}
|
||||
|
||||
\listsectionexercises
|
||||
|
||||
|
||||
%\pagebreak
|
||||
\bigskip
|
||||
\hline
|
||||
|
||||
\input{exercises.tex}
|
||||
\printcollection{banque}
|
||||
|
||||
|
||||
\end{document}
|
||||
28
1G_EnsSci/03_suites_arithmetiques/solutions.tex
Normal file
28
1G_EnsSci/03_suites_arithmetiques/solutions.tex
Normal file
@@ -0,0 +1,28 @@
|
||||
\documentclass[a4paper,10pt]{article}
|
||||
\usepackage{myXsim}
|
||||
|
||||
\usetikzlibrary{shapes.geometric}
|
||||
|
||||
\author{Benjamin Bertrand}
|
||||
\title{suites arithmétiques - Solutions}
|
||||
\tribe{1G_EnsSci}
|
||||
\date{septembre 2025}
|
||||
|
||||
\DeclareExerciseCollection{banque}
|
||||
\xsimsetup{
|
||||
exercise/print=false,
|
||||
solution/print=true,
|
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}
|
||||
|
||||
\pagestyle{empty}
|
||||
|
||||
|
||||
\begin{document}
|
||||
|
||||
\maketitle
|
||||
|
||||
\input{exercises.tex}
|
||||
%\printcollection{banque}
|
||||
%\printsolutions{exercises}
|
||||
|
||||
\end{document}
|
||||
Reference in New Issue
Block a user