1/(x+1)(x+2)+1/(x+2)(x+3)+(x+3)(x+4)+.+1/(x+2005)(x+2006)=1/2x+4012 求X
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1/(x+1)(x+2)+1/(x+2)(x+3)+(x+3)(x+4)+.+1/(x+2005)(x+2006)=1/2x+4012 求X
1/(x+1)(x+2)+1/(x+2)(x+3)+(x+3)(x+4)+.+1/(x+2005)(x+2006)=1/2x+4012 求X
1/(x+1)(x+2)+1/(x+2)(x+3)+(x+3)(x+4)+.+1/(x+2005)(x+2006)=1/2x+4012 求X
由于1/（x+i）（x+i+1）可以裂开成1/（x+i)-1/（x+i+1)
所以每一项被裂开成两部分 每项的后一部分和下一项的前一部分地抵消 只有第一项的前一部分和最后一项的后一部分没有被抵消
故原式=1/(x+1)(x+2) + 1/(x+2)(x+3) + 1/(x+3)(x+4) …… 1/(x+2005)(x+2006)=1/2x+4012
1/(x+1)-1/(x+2)+1/(x+2)-1/(x+3)……+1/(x+2005)-1/(x+2006)=1/2(x+2006)
1/(x+1)-1/(x+2006)=1/2(x+2006)
同乘2(x+1)(x+2006)
2(x+2006)-2(x+1)=x+1
2x+4012-2x-2=x+1
x+1=4010
x=4009
1/(x+1)(x+2) + 1/(x+2)(x+3) + 1/(x+3)(x+4) …… 1/(x+2005)(x+2006)=1/2x+4012
1/(x+1)-1/(x+2)+1/(x+2)-1/(x+3)……+1/(x+2005)-1/(x+2006)=1/2(x+2006)
1/(x+1)-1/(x+2006)=1/2(x+2006)
同乘2(x+1)(x+2006)
2(x+2006)-2(x+1)=x+1
2x+4012-2x-2=x+1
x+1=4010
x=4009
1/(x+1)(x+2)+1/(x+2)(x+3)+......+1/(x+2005)(x+2006)
=1/(x+1)-1/(x+2)+1/(x+2)-1/(x+3)+......+1/(x+2005)-1/(x+2006)
=1/(x+1)-1/(x+2006)
=2005/(x+1)(x+2006)
即原式化为：2005/(x+1)(x+2006)=1/2（x+2006）
解得x=4009
4009