Question

1/1X5+1/5X9+1/9X13+......+1/2005X2009

Answer 1

1/1X5+1/5X9+1/9X13+......+1/2005X2009
=(1/4)×(1-1/5+1/5-1/9+1/9-1/13+...+1/2005-1/2009)
=(1/4)×(1-1/2009)
=(1/4)×2008/2009
=502/2009

追问

1/4怎么来的

追答

1/1×5=1/5=(1/4) ×(1-1/5)
以此类推

追问

1/5X9那这个提取1/4又是怎么来的

追答

这是个规律啊。。。

1/5×9=1/45=(1/4)×4/45=(1/4)(1/5-1/9)

Answer 2

1/2005X2009=1/4(1/2005-1/2009)
你每个这样写，提取1/4相加
1/4(1-1/2009)=502/2009

追问

1/4怎么来的？

追答

公式1/(X+a)*1/(x+b)=[1/(b-a)](1/x+a-1/x+b)

Answer 3

=1/4(1-1/5+1/5-1/9******+1/2005-1/2009)
=(1-1/2009)/4
=502/2009

Answer 4

=1/4(1-1/5+1/5-1/9******+1/2005-1/2009)
=(1-1/2009)/4
=502/2009

追问

1/4怎么来的？

追答

原式可看成
1/1*(1+4)+1/(1+4)*(1+2*4)+1/(1+2*4)*(1+3*4）+………………1/(1+501*4)*(1+502*4)
=1/4(1-1/5+1/5-1/9******+1/2005-1/2009)
=(1-1/2009)/4
=502/2009

Answer 5

4（（5-1）/1x5加（9-5）/5x9……)=4x(1-1/2009)=4032/2009