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Insertion de n moyen géométrique

6ème Math-Physique - Mathématique

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Insertion de n moyen géométrique

Domaine	Science	Sous domaine	Mathématiques
Section	Scientifique	Option	Math-Physique
Discipline	Mathématique	Classe	6ème
Matériel didactique	La voie	Auteur	SCHOOLAP.COM
Objectif opérationnel	A l'issue de la leçon, l'élève sera capable de déterminer l'insertion de n moyen géométrique à l'acide de la formule de la raison géométrique en 5 minutes.
Réference	Algèbre 5e sc, cours et exercice, J.M. Makiadi, pp. 152-153.
Activité initiale
a. Rappel Déterminez trois nombres en P.G sachant que leur produit égale 3375 et leur somme égale 65 ?
b. Motivation Que faut-il faire pour trouver une P.G?		b. Motivation Pour trouver une P.G, il faut calculer sa raison.
En calculant la raison, qu'est-ce qu'on cherche ?		En calculant la raison, on insère la P.G
c. Annonce du sujet Qu'allons nous étudier aujourd'hui en math ?		c. Annonce du sujet Aujourd'hui nous allons étudier l'insertion de n moyen géométrique.
Activité principale
Que faut-il faire pour insérer n moyen géométrique ?		Pour insérer n moyen géométrique entre deux termes a et B , il suffit de calculer la raison q. Exemple : insérez trois moyens géométriques entre 2 et 200. Résolution La P.G : 2, 2 $\sqrt[]{10}, 20, 2\sqrt[]{10}, 200.$ Partagez 195 en trois parties formant une P.G sachant que la troisième partie dépasse la première de 120 ? Résolution Soient t₁, t₂, t₃ les trois parties. $\left\{ \begin{array}{rcr} t_1+t_2+t_3 & = & 195 \\ t_3=t_1+120 & & \\ \end{array} \right.$ = $\left\{ \begin{array}{rcr} t_1+t_1.t_3+t_1.q^2 & = & 195 \\ t_1.q^2=t_1+120- & & \\ \end{array} \right.$ $\left\{ \begin{array}{rcr} t_1(1+q+q^3) & = & 125 (1)\\ t_1(q^2-1)=120 (2) t_1= \frac{120}{q^2-1} & (3) & \\ \end{array} \right.$ (3) dans (1) $\frac{120}{q^2-1}(1+q+q^2)=195$ 5q²-8q-21=0 ∆=64-4(5)(-21)=64
Synthèse

Messages Timeline Exceptions Views7 Route Queries13 Models18 Gate Session Request

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