Question

a)sa se arate ca (a-b)(a+b)=a^2-b^2 oricare ar fi nr naturale a si b cu a>b. b)sa se arate ca nr N=1997+2×1998×(1+2+3+...+1996) este cub perfect.

Answer 1

a)  (a-b)(a+b)=a²+ab-ab-b²=a²-b²; ∀a,b∈N, a>b

b) N=1997+2·1998·(1+2+3+...+1996)

N=1997+2·1998·1996·1997/2

N=1997(1+1998·1996)

N=1997[1+(1997+1)(1997-1)]

N=1997(1+1997²-1²)

N=1997·1997²

N=1997³, este cub perfect