Question

1/2+1/6+1/12+1/20+......1/(2005*2006)=?

Answer 1

1/2+1/6+1/12+1/20+......1/(2005*2006)
=1-1/2+1/2-1/3+1/3-1/4+……+1/2005-1/2006
=1-1/2006
=2005/2006

Answer 2

裂项相消 这是常用数学技巧
1/6=1/(2*3)=1/2-1/3
1/12=1/(3*4)=1/3-1/4
后面规律相同
1/(2005*2006)=1/2005-1/2006
所以原式
=1/2+(1/2-1/3)+(1/3-1/4)+......+(1/2004-1/2005)+(1/2005-1/2006) 先减后加
=2005/2006
=

Answer 3

1-1/2+1/2-1/3+1/3-1/4+1/4-1/5+1/5-1/6+……+1/2005-1/2006＝2005/2006.