Question

Fie n 2 N: Aratati ca n (n + 1) (n + 2) (n + 3) + 1 este patrat
perfect.

Answer 1

$[(n+3)n][(n+1)(n+2)]=(n^2+3n)(n^2+3n+2)$.
Notam,pentru eleganta,$y=n^2+3n$.
Asadar avem $n(n+1)(n+2)(n+3)+1=y(y+2)+1=y^2+2y+1=(y+1)^2$.