1⼀1*2+1⼀2*3+1⼀3*4+1⼀4*5……+1⼀99*100=？

1/(1*2)=1-1/2
1/(2*3)=1/2-1/3
......
1/(99*100)=1/99-1/100

1/1*2+1/2*3+1/3*4+1/4*5……+1/99*100
=1-1/2+1/2-1/3+...+1/99-1/100
=1-1/100
=99/100

1/1*2+1/2*3+1/3*4+1/4*5……+1/99*100
=1-1/2+1/2-1/3+1/3-1/4+1/4-1/5+……+1/99-1/100
中间正负抵消
=1-1/100
=99/100

方法1：

公式：y=1/n(n+1) n=1、2......99
y=1/n-1/(n+1)

=1-1/2+1/2-1/3+1/3-1/4......1/99-1/100
=1-1/100
=99/100

方法2：

1/1*2+1/2*3=2/3
1/1*2+1/2*3+1/3*4=3/4
1/1*2+1/2*3+1/3*4+1/4*5=4/5
...
1/1*2+1/2*3+1/3*4+1/4*5...+1/99*100=99/100

将原式裂项后相互抵消
1/1*2+1/2*3+1/3*4+.............1/99*100
=1-1/2+1/2-1/3+1/3-1/4+...1/98-1/99+1/99-1/100
=1-1/100
=99/100

原式=1-1/100=99/100