1/x(x+1)+1/(x+1)(x+2)+1/(x+2)(x+3)+1/(x+3)(x+4)+1/(x+4)(x+5)

答：分式的裂项知识
1/x(x+1)+1/(x+1)(x+2)+1/(x+2)(x+3)+1/(x+3)(x+4)+1/(x+4)(x+5)=5/x^2+11x-708
1/x-1/(x+1)+1/(x+1)-1/(x+2)+.+1/(x+4)-1/(x+5)=5/(x^2+11x-708)
1/x-1/(x+5)=5/(x^2+11x-708)
(x+5-x)/(x^2+5x)=5/(x^2+11x-708)
5x^2+55x-3540=5x^2+10x
45x=3540
x=236/3
经检验,x=236/3是原分式方程的根

1/x(x+1)+1/(x+1)(x+2)+1/(x+2)(x+3)+1/(x+3)(x+4)
=1/x-1/(x+1)+1/(x+1)-1/(x+2),,+1/(x+3)-1/(x+4)
=1/x-1/(x+4)
=4/x(x+4)

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