Question

化简：1/x（x-1）+1/（x-1）（x-2）+1/（x-2）（x-3）+1/（x-3）（x-4）

Answer 1

1/x（x-1）+1/（x-1）（x-2）+1/（x-2）（x-3）+1/（x-3）（x-4）
＝1／﹙x－1﹚－1／x＋1／﹙x－2﹚－1／﹙x－1﹚＋1／﹙x－3﹚－1／﹙x－2﹚＋1／﹙x－4﹚－1／﹙x－3﹚
＝－1／x＋1／﹙x－4﹚
＝4／﹙x²－4﹚

Answer 2

1/x（x-1）+1/（x-1）（x-2）+1/（x-2）（x-3）+1/（x-3）（x-4）
=(x-2+x)/x(x-1)(x-2)+(x-4+x-2)/(x-2)(x-3)(x-4)
=2/x(x-2)+2/(x-2)(x-4)
=2(x-4+x)/x(x-2)(x-4)
=4/x(x-4)
=4/(x²-4x)

Answer 3

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-2)+1/(x-3)]+[-1/(x-3)+1/(x-4)]=-1/x+1/(x-4)=4/(x²-4x)

Answer 4

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

Answer 5

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