用数学归纳法证明：n(n+1)(2n+1)（n属于正实数）能被6整除.

Question

用数学归纳法证明：n(n+1)(2n+1)（n属于正实数）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．
祝愉快O(∩_∩)O~