Réponse :
a) vérification
1/3 + 1/ (2*3) = 1/3 + 1/6 = (2+1) /6 = 3/6 = 1/2 donc ok
1/4 + 1/(3*4) = 1/4 +1/12 = (3+1) /12 = 4/12 = 1/3 donc ok
1//5 + 1/ (4*5) = (4 +1)/ (4*5) = 5/ (4*5) = 1/4 donc ok
b) n est nombre entier =/ 0
1/(n+1) + 1/ (n* (n+1))= ( n +1 )/ (n* (n+1)) = 1/n
c)
1/5 = 1/6 + 1/ (5*6)
1/6 = 1/7 + 1/ (6*7)
2)
a. 2/3 = 1/2 + 1/6 = (3+1)/6 = 4/6 = 2/3
b. 2/3 = 2* (1/3) = 2* [1/4 + 1/(3*4) ]= 2* [ 1/5 + 1/ (4*5) + 1/(3*4)]
= 2* [ 1/6 + 1/ (5*6)+ 1/ (4*5) + 1/(3*4)]
= 2/6 + 2/ (5*6)+ 2/ (4*5) + 2/(3*4)
= 1/3 + 1/15 + 1/ 10 + 1/6
= 1/3 + 1/6 + 1/ 10 + 1/15
c) soit p>0 et =/ 0
1/ (2*p) + 1/(6*p) = ( 3 + 1) / 6*p = 4 / 6p = 2/ 3p
d)
2/9 = 2/ (3*3) on a p = 3
= 1/(2*3) + 1/ (6* 3) = 1/6 + 1/18
2/15 = 2/ (3*5) on p =5
= 1/ (2*5) + 1/ (6*5) = 1/10 + 1/30
3)
a )
7/11 = 14/22 = (11+2+1)/22 = 1/2 +1/11 +1/22
b)
5/7 = 10/14 = (7+2 +1)/ 14 = 1/2 +1/7 + 1/14
j' espère que cela aide
Explications étape par étape
