Question

用归纳法证明0.2+1.3+2.4+3.5+.+(n-1).(n+1) = n(n-1)(2n+5)/6

Answer 1

n=1,显然成立,略

假设n=k成立,k≥1

即0*2+1*3+……+(k-1)(k+1)=k(k-1)(2k+5)/6

则n=k+1时

0*2+1*3+……+(k-1)(k+1)+k(k+2)=k(k-1)(2k+5)/6+k(k+2)

=k(2k²+3k-5+6k+12)/6

=k(2k²+9k+7)/6

=k(k+1)(2k+7)/6

=[(k+1)-1](k+1)[2(k+1)+5]/6

所以n属于正整数时,0*2+1*3+……+(n-1)(n+1)=n(n-1)(2n+5)/6