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

来源：百度知道 编辑：UC知道 时间：2024/05/09 01:37:16
快快帮忙！！！

答案：1/(x-5)

解：
因为1/（x-1）=1/（x-1）
1/(x-1)(x-2)=1/(x-2)-1/(x-1)
1/(x-2)(x-3)=1/(x-3)-1/(x-2)
1/(x-3)(x-4)=1/(x-4)-1/(x-3)
1/(x-4)(x-5)=1/(x-5)-1/(x-4)
所以原式=1/（x-1）+1/(x-2)-1/(x-1)+1/(x-3)-1/(x-2)+1/(x-4)-1/(x-3)+1/(x-5)-1/(x-4)=1/(x-5)

裂项相消
将1/(X-1)(X-2)分为1/(X-2)-1/(X-1)后面的分为1/(X-3)-1/(X-2)约去1/(X-2)依次类推,中间项全部消去最后剩1/(X-1)+1/(X-5)(首项和尾项)
就是这样

1/x-1 +1/(x-1)(x-2)+1/(x-2)(x-3)+1/(x-3)(x-4)+1/(x-4)(x-5)
=1/[x-1]+1/[x-1]-1/[x-2]+1/[x-2]-1/[x-3]+...+1/[x-4]-1/[x-5]
=2/[x-1]-1/[x-5]
=[2x-10-x+1]/[x-1][x-5]
=[x-9]/[x-1][x-5]