问题标题：
【现在急需的数学计算~~~⑴1*2+2*3+3*4+……+100*101=⑵1*2+2*3+3*4+……+n(n+1)=⑶1*2*3+2*3*4+……+n(n+1)(n+2)=】
问题描述：
现在急需的数学计算~~~
⑴1*2+2*3+3*4+……+100*101=
⑵1*2+2*3+3*4+……+n(n+1)=
⑶1*2*3+2*3*4+……+n(n+1)(n+2)=
刘清回答：
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,欢迎交流
如果明白了,毕竟是3道题!
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