用数学归纳法证明1²+2²+3²+……+n²=n（n+1）（2n+1）/6

用数学归纳法证明1²+2²+3²+……+n²=n（n+1）（2n+1）/6
用数学归纳法证明1²+2²+3²+……+n²=n（n+1）（2n+1）/6

用数学归纳法证明1²+2²+3²+……+n²=n（n+1）（2n+1）/6
当n=1时,1*（1+1）（2*1+1）/6=1,成立.
当n=2时,2*（2+1）（2*2+1）/6=5,成立.
假设n=k时,1²+2²+3²+……+n²=n（n+1）（2n+1）/6成立,
但n=k+1时,
1²+2²+3²+……+k²+(k+1)²
=(1²+2²+3²+……+k²)+(k+1)²
=k（k+1）（2k+1）/6+(k+1)²
=(k+1)(k(2k+1)/6+(k+1))
=(k+1)((2k²+k+6k+6)/6)
=(k+1)(2k²+k+6k+6)/6
=(k+1)(2k²+7k+6)/6
=(k+1)((k+2)(2k+3))/6
=(k+1)(k+2)(2k+3)/6
=(k+1)((K+1)+1)(2(k+1)+1)/6
得证