Question

求证n(n+1)(n+2)能被6整除

Answer 1

当n=1时

n(n+1)(n+2)=6

能被6整除

设n=k时,

n(n+1)(n+2)能被6整除

即,k(k+1)(k+2)能被6整除

当n=k+1时,

(k+1)(k+2)(k+3)-k(k+1)(k+2)

=(k+1)(k+2)·[(k+3)-k]

=3(k+1)(k+2)

因为,k+1、k+2必定一奇一偶,

所以,(k+1)(k+2)是2的倍数

所以3(k+1)(k+2)是6的倍数

从而,(k+1)(k+2)(k+3)是6的倍数.

所以结论当n=k+1时也成立.

于是,

n(n+1)(n+2)能被6整除