Question

n(n+1)(2n+1)能被6整除

Answer 1

证明:

(1)当n=1时,n(n+1)(2n+1)＝1*(1+1)(2*1+1)=6

显然能被6整除

设n=k时,k(k+1)(2k+1)能被6整除

当n=k+1时,(k+1)[(k+1)+1][2(k+1)+1]

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

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

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

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

=k(k+1)(2k+1)+6(k+1)^2

由假设知k(k+1)(2k+1)+6(k+1)^2能被6整除

所以当n=k+1时,命题成立

所以原命题得证．