Patterns
Première Spécialité mathématiques
Pb-01
Un pattern est une figure géométrique (un motif) qui se répète selon une certaine logique.

Pattern 1

carre;0;0;fill:red,fill-opacity:0.2 carre;20;0;fill:red,fill-opacity:0.2 carre;30;0;fill:red,fill-opacity:0.2 carre;50;0;fill:red,fill-opacity:0.2 carre;60;0;fill:red,fill-opacity:0.2 carre;70;0;fill:red,fill-opacity:0.2

Pattern 2

carre;0;0;fill:red,fill-opacity:0.2 carre;20;0;fill:red,fill-opacity:0.2 carre;30;0;fill:red,fill-opacity:0.2 carre;50;0;fill:red,fill-opacity:0.2 carre;60;0;fill:red,fill-opacity:0.2 carre;70;0;fill:red,fill-opacity:0.2 carre;0;10;fill:red,fill-opacity:0.2 carre;20;10;fill:red,fill-opacity:0.2 carre;30;10;fill:red,fill-opacity:0.2 carre;50;10;fill:red,fill-opacity:0.2 carre;60;10;fill:red,fill-opacity:0.2 carre;70;10;fill:red,fill-opacity:0.2

Pattern 3

carre;0;0;fill:red,fill-opacity:0.2 carre;20;0;fill:red,fill-opacity:0.2 carre;20;-10;fill:red,fill-opacity:0.2 carre;30;0;fill:red,fill-opacity:0.2 carre;50;0;fill:red,fill-opacity:0.2 carre;50;-10;fill:red,fill-opacity:0.2 carre;50;-20;fill:red,fill-opacity:0.2 carre;60;0;fill:red,fill-opacity:0.2 carre;70;0;fill:red,fill-opacity:0.2

Pattern 4

carre;0;0;fill:red,fill-opacity:0.2 carre;20;0;fill:red,fill-opacity:0.2 carre;20;-10;fill:red,fill-opacity:0.2 carre;30;0;fill:red,fill-opacity:0.2 carre;30;-10;fill:red,fill-opacity:0.2 carre;50;0;fill:red,fill-opacity:0.2 carre;50;-10;fill:red,fill-opacity:0.2 carre;50;-20;fill:red,fill-opacity:0.2 carre;60;0;fill:red,fill-opacity:0.2 carre;60;-10;fill:red,fill-opacity:0.2 carre;60;-20;fill:red,fill-opacity:0.2 carre;70;0;fill:red,fill-opacity:0.2 carre;70;-10;fill:red,fill-opacity:0.2 carre;70;-20;fill:red,fill-opacity:0.2

Pattern 5

carre;0;0;fill:red,fill-opacity:0.2 carre;20;0;fill:red,fill-opacity:0.2 carre;20;-10;fill:red,fill-opacity:0.2 carre;30;0;fill:red,fill-opacity:0.2 carre;50;0;fill:red,fill-opacity:0.2 carre;50;-10;fill:red,fill-opacity:0.2 carre;50;-20;fill:red,fill-opacity:0.2 carre;60;0;fill:red,fill-opacity:0.2 carre;60;-10;fill:red,fill-opacity:0.2 carre;70;0;fill:red,fill-opacity:0.2

Pattern 6

carre;0;0;fill:red,fill-opacity:0.2 carre;-10;10;fill:red,fill-opacity:0.2 carre;10;-10;fill:red,fill-opacity:0.2 carre;60;0;fill:red,fill-opacity:0.2 carre;50;10;fill:red,fill-opacity:0.2 carre;40;20;fill:red,fill-opacity:0.2 carre;70;-10;fill:red,fill-opacity:0.2 carre;80;-10;fill:red,fill-opacity:0.2 carre;70;-20;fill:red,fill-opacity:0.2 carre;130;0;fill:red,fill-opacity:0.2 carre;120;10;fill:red,fill-opacity:0.2 carre;110;20;fill:red,fill-opacity:0.2 carre;100;30;fill:red,fill-opacity:0.2 carre;140;-10;fill:red,fill-opacity:0.2 carre;150;-10;fill:red,fill-opacity:0.2 carre;160;-10;fill:red,fill-opacity:0.2 carre;140;-20;fill:red,fill-opacity:0.2 carre;140;-30;fill:red,fill-opacity:0.2

Pattern 7

carre;0;0;fill:red,fill-opacity:0.2 carre;10;0;fill:red,fill-opacity:0.2 carre;40;0;fill:red,fill-opacity:0.2 carre;50;0;fill:red,fill-opacity:0.2 carre;60;0;fill:red,fill-opacity:0.2 carre;40;-10;fill:red,fill-opacity:0.2 carre;50;-10;fill:red,fill-opacity:0.2 carre;60;-10;fill:red,fill-opacity:0.2 carre;90;0;fill:red,fill-opacity:0.2 carre;100;0;fill:red,fill-opacity:0.2 carre;110;0;fill:red,fill-opacity:0.2 carre;120;0;fill:red,fill-opacity:0.2 carre;90;-10;fill:red,fill-opacity:0.2 carre;100;-10;fill:red,fill-opacity:0.2 carre;110;-10;fill:red,fill-opacity:0.2 carre;120;-10;fill:red,fill-opacity:0.2 carre;90;-20;fill:red,fill-opacity:0.2 carre;100;-20;fill:red,fill-opacity:0.2 carre;110;-20;fill:red,fill-opacity:0.2 carre;120;-20;fill:red,fill-opacity:0.2

Pattern 8

carre;0;0;fill:red,fill-opacity:0.2 carre;-10;-10;fill:red,fill-opacity:0.2 carre;-10;-20;fill:red,fill-opacity:0.2 carre;10;-10;fill:red,fill-opacity:0.2 carre;50;0;fill:red,fill-opacity:0.2 carre;60;0;fill:red,fill-opacity:0.2 carre;40;10;fill:red,fill-opacity:0.2 carre;40;-10;fill:red,fill-opacity:0.2 carre;40;-20;fill:red,fill-opacity:0.2 carre;40;-30;fill:red,fill-opacity:0.2 carre;70;-10;fill:red,fill-opacity:0.2 carre;80;-10;fill:red,fill-opacity:0.2 carre;70;-20;fill:red,fill-opacity:0.2 carre;80;-20;fill:red,fill-opacity:0.2 carre;120;0;fill:red,fill-opacity:0.2 carre;130;0;fill:red,fill-opacity:0.2 carre;140;0;fill:red,fill-opacity:0.2 carre;110;10;fill:red,fill-opacity:0.2 carre;100;20;fill:red,fill-opacity:0.2 carre;110;-10;fill:red,fill-opacity:0.2 carre;110;-20;fill:red,fill-opacity:0.2 carre;110;-30;fill:red,fill-opacity:0.2 carre;110;-40;fill:red,fill-opacity:0.2 carre;150;-10;fill:red,fill-opacity:0.2 carre;160;-10;fill:red,fill-opacity:0.2 carre;170;-10;fill:red,fill-opacity:0.2 carre;150;-20;fill:red,fill-opacity:0.2 carre;160;-20;fill:red,fill-opacity:0.2 carre;170;-20;fill:red,fill-opacity:0.2 carre;150;-30;fill:red,fill-opacity:0.2 carre;160;-30;fill:red,fill-opacity:0.2 carre;170;-30;fill:red,fill-opacity:0.2

Pattern 9

carre;0;0;fill:red,fill-opacity:0.2 carre;10;0;fill:red,fill-opacity:0.2 carre;20;0;fill:red,fill-opacity:0.2 carre;0;-10;fill:red,fill-opacity:0.2 carre;10;-10;fill:red,fill-opacity:0.2 carre;20;-10;fill:red,fill-opacity:0.2 carre;0;-20;fill:red,fill-opacity:0.2 carre;10;-20;fill:red,fill-opacity:0.2 carre;20;-20;fill:red,fill-opacity:0.2 carre;10;-30;fill:red,fill-opacity:0.2 carre;50;0;fill:red,fill-opacity:0.2 carre;60;0;fill:red,fill-opacity:0.2 carre;70;0;fill:red,fill-opacity:0.2 carre;80;0;fill:red,fill-opacity:0.2 carre;50;-10;fill:red,fill-opacity:0.2 carre;60;-10;fill:red,fill-opacity:0.2 carre;70;-10;fill:red,fill-opacity:0.2 carre;80;-10;fill:red,fill-opacity:0.2 carre;50;-20;fill:red,fill-opacity:0.2 carre;60;-20;fill:red,fill-opacity:0.2 carre;70;-20;fill:red,fill-opacity:0.2 carre;80;-20;fill:red,fill-opacity:0.2 carre;60;-30;fill:red,fill-opacity:0.2 carre;70;-30;fill:red,fill-opacity:0.2 carre;60;-40;fill:red,fill-opacity:0.2 carre;110;0;fill:red,fill-opacity:0.2 carre;120;0;fill:red,fill-opacity:0.2 carre;130;0;fill:red,fill-opacity:0.2 carre;140;0;fill:red,fill-opacity:0.2 carre;150;0;fill:red,fill-opacity:0.2 carre;110;-10;fill:red,fill-opacity:0.2 carre;120;-10;fill:red,fill-opacity:0.2 carre;130;-10;fill:red,fill-opacity:0.2 carre;140;-10;fill:red,fill-opacity:0.2 carre;150;-10;fill:red,fill-opacity:0.2 carre;110;-20;fill:red,fill-opacity:0.2 carre;120;-20;fill:red,fill-opacity:0.2 carre;130;-20;fill:red,fill-opacity:0.2 carre;140;-20;fill:red,fill-opacity:0.2 carre;150;-20;fill:red,fill-opacity:0.2 carre;120;-30;fill:red,fill-opacity:0.2 carre;130;-30;fill:red,fill-opacity:0.2 carre;140;-30;fill:red,fill-opacity:0.2 carre;120;-40;fill:red,fill-opacity:0.2 carre;130;-40;fill:red,fill-opacity:0.2

Pattern 10

carre;0;0;fill:red,fill-opacity:0.2 carre;10;0;fill:red,fill-opacity:0.2 carre;40;0;fill:red,fill-opacity:0.2 carre;50;0;fill:red,fill-opacity:0.2 carre;40;-10;fill:red,fill-opacity:0.2 carre;50;-10;fill:red,fill-opacity:0.2 carre;80;0;fill:red,fill-opacity:0.2 carre;90;0;fill:red,fill-opacity:0.2 carre;100;0;fill:red,fill-opacity:0.2 carre;110;0;fill:red,fill-opacity:0.2 carre;80;-10;fill:red,fill-opacity:0.2 carre;90;-10;fill:red,fill-opacity:0.2 carre;100;-10;fill:red,fill-opacity:0.2 carre;110;-10;fill:red,fill-opacity:0.2 carre;140;0;fill:red,fill-opacity:0.2 carre;150;0;fill:red,fill-opacity:0.2 carre;160;0;fill:red,fill-opacity:0.2 carre;170;0;fill:red,fill-opacity:0.2 carre;140;-10;fill:red,fill-opacity:0.2 carre;150;-10;fill:red,fill-opacity:0.2 carre;160;-10;fill:red,fill-opacity:0.2 carre;170;-10;fill:red,fill-opacity:0.2 carre;140;-20;fill:red,fill-opacity:0.2 carre;150;-20;fill:red,fill-opacity:0.2 carre;160;-20;fill:red,fill-opacity:0.2 carre;170;-20;fill:red,fill-opacity:0.2 carre;140;-30;fill:red,fill-opacity:0.2 carre;150;-30;fill:red,fill-opacity:0.2 carre;160;-30;fill:red,fill-opacity:0.2 carre;170;-30;fill:red,fill-opacity:0.2
Travail de recherche.
  1. Pour chaque pattern dessinez les deux motifs suivants.
  2. Calculez le nombre total de carrés nécessaires pour les motifs de rang 1 à 5.
  3. Déterminez une formule permettant de calculer le nombre de carrés en fonction du rang du motif. Justifiez.
  4. Déterminez une formule permettant de calculer la somme des nombres de carrés des motifs de rang 1 à n. Justifiez.
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