Question

1/1*2+1/2*3+1/3*4++1/(X+1)(x+2)= 2011/2012

Answer 1

解：
注意到：
1/1*2 = (2-1)/1*2=1/1 - 1/2
1/2*3 = (3-2)/2*3=1/2 - 1/3
1/3*4 = (4-3)/3*4=1/3 - 1/4
......
1/(X+1)(x+2) = 1/(x+1) - 1/(x+2)
因此：上述各式相加，则：
左边=1/1*2+1/2*3+1/3*4+.....+1/(X+1)(x+2)
右边=1/1 - 1/2+1/2 - 1/3+1/3 - 1/4+1/(x+1) - 1/(x+2)
=1-1/(x+2)
因此，原方程为：
1-1/(x+2) = 2011/2012
即：
(x+1)/(x+2) = 2011/2012
∴
2012(x+1)=2011(x+2)
2012x+2012=2011x+4022
x=2010

Answer 2

1/1*2=1-1/2

1/2*3=1/2-1/3
1/3*4=1/3-1/4
。。。
。。。
1/(x+1)(x+2)=1/(x+1)-1/(x+2)
所以，左式=1-1/2+1/2-1/3+1/3-1/4+。。。+1/(x+1)-1/(x+2)
=1-1/(x+2)
=(x+1)/(x+2)
所以，(x+1)/(x+2)=2011/2012
得：x=2010

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Answer 3

1/1*2+1/2*3+1/3*4++1/(X+1)(x+2)= 2011/2012
1-1/2+1/2-1/3+……+1/(X+1)-1/(x+2)= 2011/2012
1-1/(x+2)= 2011/2012
1/(x+2)=1- 2011/2012
1/(x+2)=1/2012
x+2=2012
x=2010