Question

用数学归纳法证明2²+4²+6²+…+(2n)²=n(n+1)(2n+1).

      

Answer 1

证明:(1)当n=1时,左边=2²=4,右边=×1×2×3=4,∴命题成立.?

       (2)假设当n=k时命题成立,即2²+4²+6²+…+(2k)²=k(k+1)(2k+1),?

       那么当n=k+1时,2²+4²+6²+…+(2k)²+(2k+2)²??

       =(2k+2)²+k(k+1)(2k+1)?

       = (k+1)［(4k+4)+k(2k+1)］?

       = (k+1)(2k²+7k+6)?

       = (k+1)(k+2)(2k+3)?

       = (k+1)［(k+1)+1］［2(k+1)+1］,?

       即当n=k+1时,命题也成立.?

       由(1)(2)可知,命题对所有n∈N^*都成立.