Question

计算x(x+1)分之1+（x+1)(x+2)分之1+（x+2)(x+3)分之1+……+（x+2011)(x+2012)分之1

Answer 1

x(x+1)分之1+（x+1)(x+2)分之1+（x+2)(x+3)分之1+……+（x+2011)(x+2012)分之尺袭慎1
=1/x-1/(x+1）+1/（x+1)-1/(x+2)+1/（x+2)-1/(x+3)+…陵敬禅唤…+1/（x+2011)-1/(x+2012)
=1/x-1/(x+2012)
=2012/[x(x+2012)]

Answer 2

裂激陪项相消：隐铅裂
1/[x(x+1)]+1/[(x+1)(x+2)]+1/[(x+2)(x+3)]+...+1/[(x+2011)(x+2012)]
=1/x-1/(x+1)+1/(x+1)-1/(x+2)+1/(x+2)-1/(x+3)+...+1/(x+2011)-1/灶闭(x+2012) 中间的都消掉了
=1/x-1/(x+2012)
=(x+2012-x)/[x(x+2012)]
=2012/[x(x+2012)]