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APPLIQUER LES IDENTITES REMARQUABLES
Avec l'aimable autorisation des ditions Bordas Myriade 3e
Commentaires:
Ces quatre problmes, plutt ouverts, permettent de mettre en application les identits remarquables.
L'activit se prte bien un travail en groupe en donnant la possibilit aux lves d'effectuer les problmes dans le dsordre.
PROBLEME 1
1. Effectuer les calculs ci-dessous :
a. 1232 1222 1212 + 1202b. 452 442 432 + 422
c. 872 862 852 + 842Quelle remarque peut-on faire concernant les rsultats ?2. Choisir quatre nombres conscutifs et effectuer les mmes calculs qu la question 1.3. A laide des questions prcdentes, crire une conjecture.
4. Expliquer pourquoi la conjecture peut scrire ainsi :
(n + 3)2 (n + 2)2 (n + 1)2 + n2 = 4.5. Prouver que cette galit est vraie pour tout nombre n entier et conclure.
PROBLEME 2 Prove what you think !
Kevin notes that :52 42 = 5 + 4 ; 102 92 = 10 + 9 ; 2502 2492 = 250 + 249 and thus, he asserts that the difference between the squares of two consecutive numbers is equal to the sum of this two consecutive numbers.Mary tells him his conjecture couldnt be right for any consecutive numbers. How can Dawson prove hes right ?
PROBLEME 3
Voici un programme de calcul :
1. Ecrire lexpression finale obtenue si lon prend x comme nombre de dpart.2. M o n t r e r q u e c e t t e e x p r e s s i o n e s t g a l e a 1 6 x 2 " 6 4 x + 6 4 .
P R O B L E M E 4
O n c o n s i d r e l e s n o m b r e s s u i v a n t s : A = 1 0 0 1 9 9 9 " 9 9 9 2 , B = 5 7 5 5 " 5 5 2 e t C = ( "2 ) ( "4 ) " ( "4 ) 2 . 1 . D o n n e r l e s v a l e u r s l u e s s u r l a c a l c u l a t r i c e p o u r A , B e t C .
2 . O n p o s e D = ( x + 1 ) ( x " 1 ) " ( x " 1 ) 2 . x t a n t u n n o m b r e e n t i e r , s u p r i e u r a 1 , m o n t r e r q u e D e s t u n m u l t i p l e d e 2 . 3 . T r o u v e r u n e e x p r e s s i o n E d e l a m m e f o r m e q u e c e l l e d e A p o u r l a q u e l l e l e r s u l t a t d u c a l c u l e s t 2 0 0 8 .
B r e v e t M a d a g a s c a r , 2 0 0 8
Y v a n M o n k a A c a dmie de Strasbourg HYPERLINK "http://www.maths-et-tiques.fr" www.maths-et-tiques.fr
Hors du cadre de la classe, aucune reproduction, mme partielle, autres que celles prvues l'article L 122-5 du code de la proprit intellectuelle, ne peut tre faite de ce site sans l'autorisation expresse de l'auteur.
HYPERLINK "http://www.maths-et-tiques.fr/index.php/mentions-legales" www.maths-et-tiques.fr/index.php/mentions-legales
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