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

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