解方程(x/x(x+2))+(x/(x+2)(x+4))+......+x/(x+8)(x+10)=1

来源：百度知道 编辑：UC知道 时间：2024/09/20 21:28:44
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(x/x(x+2))+(x/(x+2)(x+4))+......+x/(x+8)(x+10)
=(x/2)*(2/x(x+2))+(2/(x+2)(x+4))+......+2/(x+8)(x+10)
=(x/2)*[1/x-1/(x+2)+1/(x+2)-1/(x+4)+......+1/(x+8)-1/(x+10)]
=(x/2)*[1/x-1/(x+10)]
=1/2-x/2(x+10)=1
x/2(x+10)=-1/2
-x=x+10
x=-5
分式方程要检验
经检验，x=-5是方程的解

=x*[(1/x(x+2))+(1/(x+2)(x+4))+......+(1/(x+8)(x+10))]
对方括号内裂项相消得
=x*1/2*(1/x-1/(x+10))=1
故x/(x+10)=-1,x=-5

原式=(x/2)*(2/x(x+2))+(2/(x+2)(x+4))+......+2/(x+8)(x+10)
=(x/2)*[1/x-1/(x+2)+1/(x+2)-1/(x+4)+......+1/(x+8)-1/(x+10)]
=(x/2)*[1/x-1/(x+10)]
=1/2-x/2(x+10)=1
x/2(x+10)=-1/2
-x=x+10
x=-5