Question

(1*3)分之一+(3*5)分之一+(5*7)分之一+……+(2007*2003)分之一

Answer 1

（1*3)分之一+(3*5)分之一+(5*7)分之一+……+(2007*2009)分之一
=（1-1/3）/2+(1/3-1/5)/2+(1/5-1/7)/2+……+
(1/2007-1/2009)/2
=(1-1/3+1/3-1/5+1/5-1/7+……+1/2007-1/2009)/2
=(1-1/2009)/2
=1004/2009

Answer 2

原式最后应该是2001*2003吧
原式=1/2（（1-1/3）+（1/3-1/5）+......+（1/2001
-1/2003）)=1/2(1-1/2003)=2001/2003

Answer 3

1/(1*3)+1/(3*5)+...+1/(2007*2009)
=1/2*(1/1-1/3)+1/2*(1/3-1/5)+...+1/2*(1/2007-1/2009)
=1/2*[(1-1/3)+(1/3-1/5)+...+(1/2007-1/2009)]
=1/2*(1-1/2009)
=(1/2)*(2008/2009)
=1004/2009.