1乘2+2乘3+3乘4+4乘5+···+99乘100

题目

1乘2+2乘3+3乘4+4乘5+···+99乘100

答案

这个问题,我相当熟悉了,分析出来,你认真看吧.
1X2 +2X3 +3X4
= (1X2X3 +2X3X3 +3X4X3) /3
= [ 1X2X3 +2X3X(4-1) +3X4X(5-2) ] /3
= [ 1X2X3 -1X2X3 +2X3X4 -2X3X4 +3X4X5 ] /3
= 3X4X5 /3
1X2 +2X3 +3X4 +4X5
= (1X2X3 +2X3X3 +3X4X3 +4X5X3) /3
= [ 1X2X3 +2X3X(4-1) +3X4X(5-2) +4X5X(6-3) ] /3
= [ 1X2X3 -1X2X3 +2X3X4 -2X3X4 +3X4X5 -3X4X5 +4X5X6 ] /3
= 4X5X6 /3
规律你看出来了吗?这个数列的公式就是
通项 a= n(n+1)
前n项数列和 S= n(n+1)(n+2)/3
这样一来,一直加到 99X100,就是
1X2 +2X3 +3X4 +4X5 +……+99X100
= 99X100X101 /3
= 333300