1*2+2*3+3*4+4*5+.+29*30

1*2+2*3+3*4+4*5+.+29*30
1*2+2*3+3*4+4*5+.+29*30

1*2+2*3+3*4+4*5+.+29*30
1+2+……+n=n(n+1)/2
1^2+2^2+……+n^2=n(n+1)(2n+1)/6
n(n+1)=n^2+n
所以1*2+2*3+……+n(n+1)=n(n+1)/2+n(n+1)(2n+1)/6
=n(n+1)[1/2+(2n+1)/6]
=n(n+1)(3+2n+1)/6
=n(n+1)(n+2)/3
所以1*2+2*3+3*4+4*5+.+29*30
此时n=29
=29*30*31/3
=(30-1)*(30+1)*30/3
=(30^2-1)*10
=(900-1)*10
=8990