n(n+1)=(n(n+1)(n+2)-(n-1)n(n+1))/3
⑵1*2+2*3+3*4+……+n(n+1)
=(1*2*3-0*1*2)/3+(2*3*4-1*2*3)/3+(3*4*5-2*3*4)/3+...+(n(n+1)(n+2)-(n-1)n(n+1))/3
=(n(n+1)(n+2)-0*1*2)/3
=n(n+1)(n+2)/3
(1)把n=100代入上式,得:
1*2+2*3+3*4+···100*101
=100*(100+1)*(100+2)/3
=100*101*102/3
=343400
(3)
n(n+1)(n+2)=(n(n+1)(n+2)(n+3)-(n-1)n(n+1)(n+2))/4
1*2*3+2*3*4+……+n(n+1)(n+2)
=(1*2*3*4-0*1*2*3)/4+(2*3*4*5-1*2*3*4)/4+...+(n(n+1)(n+2)(n+3)-(n-1)n(n+1)(n+2))/4
=(n(n+1)(n+2)(n+3)-0*1*2*3)/4
=n(n+1)(n+2)(n+3)/4
------若有不明白,歡迎交流
如果明白了,畢竟是3道題!