Question

1/1*2*3+1/2*3*4+......+1/98*99*100

1乘2乘3分之一加2乘3乘4分之一按规律加到98乘99乘100分之一。求和。

Answer 1

1/n(n+1)(n+2) = 1/(n+1) [1/2(1/n-1/(n+2))] = 1/2[(1/n - 1/(n+1)) -(1/(n+1) -1/(n+2) ]
= 1/2[(1/n - 2/(n+1)) + 1/(n+2)]

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

=1-1/2-1/99+1/100
=4949/9900