Question

解方程：1/x(x+2)+1/(x+2)(x+4)+……+1/(x+8)(x+10)=5/24

1/x(x+2)+1/(x+2)(x+4)+……+1/(x+8)(x+10)=5/24

Answer 1

1/x(x+2)+1/(x+2)(x+4)+……+1/(x+8)(x+10)=5/24
1/(2x)-1/{2(x+2)) + 1/{2(x+2)}-1/{2(x+4)} +……+1/{2(x+8)} - 1/{2(x+10)} = 5/24
1/(2x)-1/{2(x+10)} = 5/24
1/x-1/(x+10) = 5/12
10/{x(x+10)} = 5/12
2/{x(x+10)} =1 /12
x(x+10)=24
x^2+10x-24=0
(x+12)(x-2)=0
分母中不含(x+12)和(x-2)
x1=-12，x2=2

Answer 2

1/x-1/(x+2)=2/x(x+2)
则：1/x(x+2)=1/2[1/x-1/(x+2)]
同理，其它项也可以如此变形，所以：
1/x(x+2)+1/(x+2)(x+4)+……+1/(x+8)(x+10)
=1/2[1/x-1/(x+2)+1/(x+2)-1/(x+4)+......+1/(x+6)-1/(x+8)+1/(x+8)-1/(x+10)]
=1/2[1/x-1/(x+10)]
=5/24
要解的就是1/2[1/x-1/(x+10)]=5/24
10/x(x+10)=5/12
(x+12)(x-2)=0
x=-12或x=2

Answer 3

x=-2或12
1/x(x+2)=(1/x-1/(x+2))/2

Answer 4

1/2x-1/2(x+2)+1/2(x+2)-1/2(x+4)+……+1/2(x+8)-1/2(x+10)=5/24
1/2x-1/2(x+10)=5/24
10/x(x+10)=5/12
x=2或x=-12