F.A.Q. DE WEBMATHS

Maths » Lycée » Analyse
Comment prouver qu'une suite est une suite géométrique et en donner la raison et le premier terme ?

Le problème

On donne une suite définie par un problème de croissance, reconnaître si cette suite est géométrique et dans ce cas en donner la raison et le premier terme.
Les suites géométriques concernent tous les phénomènes économique comme les croissances cumulée en %, comme par exemple les intérets composés d'emprunts.

La propriétés.

Ce sont les propriétés découlant de la définition d'une suite géométrique :
Une suite géométrique est une suite dont chacun des termes est obtenu en multipliant par un nombre constant le terme précédent précédent.

$u_{n+1} = u_n \times q$ où q est la raison de la suite.

Propriété

Si $\frac{u_{n+1}}{ u_n} =q$ alors la suite est une suite géométrique.

Propriété

Et le calcul du n ième terme connaissant le premier et la raison.
$u_n = u_0\times q^n$

La méthode.

On exprime le terme de rang n et celui de rang n+1. On fait la division du second par le premier : $\frac{u_{n+1}}{ u_n}$ . Si on trouve un nombre constant (c'est à dire qui n'est pas fonction de n) alors la suite est géométrique et le nombre trouvé en est la raison.

L'exemple.

La population de bactéries d'une culture augmente de 4% toutes les heures. On a fait un comptage à 12 h, il y avait 50 000 bactérie en culture.
Montrer que la population des bactéries augmente selon une suite géométrique dont on déterminera la raison.

Solution

On calcule la population de bactéries au bout d'une heure : $b_1= 50 000 + 50 000\times 0.04 = 50 000 \times 1.04$
Au bout de deux heures, on aura $b_2 = b_1 \times 1,04 = 50 000 \times 1,04^2$
Au bout de n heures on aura : $b_n = 50 000 \times 1.04^n$ et $b_{n+1} = 1.04b_n$

Le rapport $\frac{b_{n+1}}{b_n}=1.04$ On a bien une suite géométrique de raison 1.04 et de premier terme est 50 000.

Last update: 2006-02-08 19:11
Author: Henry Serge

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Comment of camille:
la fiche est simple, précise et par là claire, il est intéresant d'y avoir ajouté un cas pratique puisqu'on peut ainsi vérifier si l'on a bien compris la fiche et observer la démonstration ainsi que la rédaction. je crois pourtant qu'il y a une erreur dans la propriété, il est en effet noté "une suite arithmétique..." alors que c'est la définition d'une suite géométrique, mais peut etre est-ce normal...?

Comment of Matthieu:
Je me permets d'étayer la remarque de Camille, il y a bien une faute dans la propriété : il s'agit bel et bien d'une SUITE GEOMETRIQUE et non arithmétique comme inscrit.
Cependant, cet article est très utile et correctement synthétisé.
Bravo !

Comment of Dahevos:
Il y a une petite faute ( a régler mais c'est très léger ).

C'est la répétition du mot précédant dans cette mini-fiche.
Voila ^^

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Maths » Lycée » Analyse Comment prouver qu'une suite est une suite géométrique et en donner la raison et le premier terme ?

Le problème

La propriétés.

Propriété

Propriété

La méthode.

L'exemple.

Maths » Lycée » Analyse
Comment prouver qu'une suite est une suite géométrique et en donner la raison et le premier terme ?