用数学归纳法证明n(n+1)(n+2)能被3整除

题目

用数学归纳法证明n(n+1)(n+2)能被3整除

答案

n=1,n(n+1)(n+2)=1*2*3=6,显然成立
假设n=k时,k(k+1)(k+2)能被3整除
当n=k+1时,
n(n+1)(n+2)
=（k+1)(k+2)(k+3)
=k(k+1)(k+2)+3(k+1)(k+2),
由假设知：式中第一项k(k+1)(k+2)能被3整除,
第二项3(k+1)(k+2)也能被3整除
所以当n=k+1时,
n(n+1)(n+2)
=（k+1)(k+2)(k+3)能被3整除
综上可知,n(n+1)(n+2)能被3整除