bjr
a)
(x - 1) (x⁵ + x⁴ + x³+ x² + x + 1) =
x⁶ + x⁵ + x⁴ + x³+ x² + x - x⁵ - x⁴ - x³- x² - x - 1) =
x⁶ - 1
( l'énoncé n'est pas correct)
b)
1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 = 1 + (1/2) + (1/2)² + (1/2)³ + (1/2)⁴ + (1/2)⁵
on utilise le résultat de la première question
x⁶ - 1 = (x - 1) (x⁵ + x⁴ + x³+ x² + x + 1)
d'où
(x⁵ + x⁴ + x³+ x² + x + 1) = (x⁶ - 1)/(x - 1)
ici x = 1/2
1 + (1/2) + (1/2)² + (1/2)³ + (1/2)⁴ + (1/2)⁵ = [(1/2)⁶ - 1] / (1/2 - 1)
= [(1/64) - 1]/(-1/2)
= -(63/64) / (-1/2)
= (63/64) * 2 =
= 63/32