1/x(x+1)+1/(x+1)(x+2)+1/(x+2)(x+3)+.+1/(x+999)(x+1000)
问题描述：
1/x(x+1)+1/(x+1)(x+2)+1/(x+2)(x+3)+.+1/(x+999)(x+1000)
答
1/[x(x+1)]+1/[(x+1)(x+2)]+1/[(x+2)(x+3)]+......+1/[(x+999)(x+1000)]
=[1/x-1/(x+1)]+[1/(x+1)-1/(x+2)]+[1/(x+2)-1/(x+3)]+……+[1/(x+999)-1/(x+1000)]
=1/x-1/(x+1)+1/(x+1)-1/(x+2)+1/(x+2)-1/(x+3)+……+1/(x+999)-1/(x+1000)
=1/x-1/(x+1000)
=[(x+1000)-x]/[x(x+1000)]
=1000/[x(x+1000)]
答
原式=1/x-1/(x+1)+1/(x+1)-1/(x+2)+1/(x+2)-1/(x+3)+......+1/(x+999)-1/(x+1000)
=1-1/(x+1000)
=(x+999)/(x+1000)
答
1/x(x+1)=1/x-1/(x+1)
1/(x+1)(x+2)=1/(x+1)-1/(x+2)
...
1/x(x+1)+1/(x+1)(x+2)+1/(x+2)(x+3)+.+1/(x+999)(x+1000)
=1/x-1/(x+1)+1/(x+1)-1/(x+2)+...+1/(x+999)-1/(x+1000)
=1/x-1/(x+1000)=1000/x(x+1000)