Question

初一分式题

1/x(x+1)+1/(x+1)(x+2) +1/(x+2)(x+3)+.......+1/(x+2099)(x+2100)

求过程

Answer 1

先算1/x(x+1)+1/(x+1)(x+2)=2/x(x+2)
1/x(x+1)+1/(x+1)(x+2) +1/(x+2)(x+3)=3/x(x+3)
1/x(x+1)+1/(x+1)(x+2) +1/(x+2)(x+3)+1/(x+3)(x+4)=4/x(x+4)
由上可得出：
1/x(x+1)+1/(x+1)(x+2) +1/(x+2)(x+3)+.......+1/(x+2099)(x+2100)=2100/(x+2099)(x+2100)

Answer 2

1/x(x+1)+1/(x+1)(x+2) +1/(x+2)(x+3)+.......+1/(x+2099)(x+2100)
=1/x-1/(x+1)+1/(x+1)-1/(x+2)+1/(x+2)-1/(x+3)+...+1/(x+2099)-1/(x+2100)
=1/x -1/(x+2100)
=2100/x(x+2100)