1x2+2x3+3x4+4x5........99x100

1x2+2x3+3x4+4x5........99x100

=1/3*1*2*3+1/3*[2*3*4-1*2*3]+1/3[3*4*5-2*3*4]+...+1/3[99*100*101-98*99*100]

=1/3[1*2*3+2*3*4-1*2*3+3*4*5-2*3*4+....+99*100*101-98*99*100]

=1/3*99*100*101

=3300*101

=333300

公式：Sn=1*2+2*3+3*4+...+n(n+1)=n(n+1)(n+2)/3

最简洁明了的应是：
∑(k-1)k=∑k^2 - ∑k=(n-1)n(n+1)/3,(k=1,2,3...n)

把100代入上式

1x2+2x3+3x4+4x5+5x6+...+99x100=99*100*101/3= 333300

100(1*100)/2+2(1-3/2)/1-3/2