已知1/1*2=1-1/2,1/2*3=1/2-1/3,1/3*4=1/3-1/4.

你说的规律
(1):1/n*(n+1)=1/n -1/n+1
(2)(a+b)/a*b= 1/a + 1/b
这都是显然得很的.
原式=2(x+2)/(x+1)(x+3) -2(x+4)/(x+3)(x+5)+2(x+6)/(x+5)(x+7)-2(x+8)/(x+7)(x+9)
=(x+2)/(x+1)-(x+2)/(x+3)-(x+4)/(x+3)+(x+4)/(x+5)+...-(x+8)/(x+7)+(x+8)/(x+9)
=(x+2)/(x+1)-2+2-2+(x+8)/(x+9)
=1/(x+1)-1/(x+9)
=8/(x^2+10+9)