Question

用数学归纳法证明命题n+(n+1)+...+2n=3n(n+1)/2时,在作了归纳假设后,需要证明当n=k+1时

Answer 1

1>n=1时,左边=3,右边=3,成立.

2>n=k成立时,即k+(k+1)+.+2k=3k(k+1)/2,

则当n=k+1时,

(k+1)+(k+2)+.+2k+（2k+1)+(2k+2)

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

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

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

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

故当n=k+1时同样成立

得证.