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——=_NextPart_01DBB65E.09E4C410–
=
1. =
Par un calcul simple (
)
Il faut d’abord
connaître la définition : une suite 
est croissante si et seulement si&n=
bsp;
. Une suite est décroissante si et seule=
ment
si 
L’exercice 1 de la
session 2004 offre cette possibilité. 
.
Il suffit donc de calcul=
er 
. Ce calcul montre alors que la suite=
est croissante. Au passage, quand on
calcule , si le résultat est positif o=
u nul,
la suite est croissante, par contre si le résultat est négati=
f ou
nul, la suite est décroissante. Enfin si le résultat est
indépendant de n, alors la suite est arithmétique
=
2. =
Raisonnement par récurrence
Exemple : Soit une suite définie de manière
récurrente par : <=
![if !msEquation]>
avec 
.
Soit alors
. Il est clair que
, et donc f est croissante sur 
.
Soit P(n) la propriété&=
nbsp;:
.
Montrons par récurrence que est croissante
1° pas : P(0) est vraie ca=
r et 
; donc 
2° pas : Hypothèse de
récurrence :
 =
; Soit
n quelconque. Supposons que P(n-1) est vraie soit :
3° pas : Montrons que P(n)=
est
vraie
 =
; D’après
l’hypothèse de récurrence on a
Or f est croissante sur 
, donc 
et par définition . On vient de montrer que P(n-1) impl=
ique
P(n) ; grâce au raisonnement par récurrence on vient donc=
de
montrer que , c’est à dire que la su=
ite est croissante.
PS : En toute rigueur, il aurait=
fallu
montrer que 
, car en appliquant f à , on ne sait pas s’il se pose un
problème d’existence ;
Attention
, si f est croissante, on peut seulem=
ent
dire que la suite est monotone. Tout dépend de=
s 2
premiers termes. S’ils sont en ordre croissant, la suite sera croissa=
nte,
s’ils sont en ordre décroissant, la suite sera
décroissante.
3. =
Par calcul de 
 =
; En
effet, si le résultat est supérieur ou égal à 1=
, la
suite est croissante, par contre si le résultat est inférieur=
ou
égal à 1, la suite est décroissante. Enfin, si le
résultat est indépendant de n, la suite est
géométrique.
 =
;
 =
;