Question

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

Answer 1

1/铅橘行n(n+1)(n+2)=[1/n(n+1)-1/(n+1)(n+2)]/2
1/1x2x3 + 1/2x3x4 +1/3x4x5 + … + 1/n(n+1)(n+2)
=[和1/n(n+1)-和1/(n+1)(n+2)]/2
=[1-1/槐哗(n+1)-1/2+1/(n+2)]/2
=1/4-1/2(n+1)(n+2)

1/n(n+2)=[1/n-1/(n+2)]/2
1/1x3 + 1/2x4 + 1/3x5 … + 1/n(n+2)
=[1+1/2-1/(n+1)-1/(n+2)]/2
=3/4-1/2(n+1)-1/伍派2(n+2)
把数带入即可
3/4-1/2*99-1/2*100

Answer 2

1/1*2*3=1/2(1/1*2-1/2*3)
1/98*99*100=1/2(1/98*99-1/99*100)
1/1*2*3+1/2*3*4+........+1/98*99*100
=1/拦游旅2(1/1*2-1/磨仿2*3+1/2*3-1/3*4+……简凳+1/97*98-1/98*99+1/98*99-1/99*100)
=1/2(1/1*2-1/99*100)
=4949/19800