Question

1/a(a+1) + 1/(a+1)(a+2) + ...+ 1/(a+2007)(a+2008)

Answer 1

1/a(a+1)+1/(a+1)(a+2)+1/(a+2)(a+3).+1(a+2007)(a+2008)

第一项1/a(a+1)=1/a-1/(a+1)

第二项1/(a+1)(a+2)=1/(a+1)-1/(a+2)

第三项1/(a+2)(a+3)=1/(a+2)-1/(a+3)

.

第2008项1/(a+2007)(a+2008)=1/(a+2007)-1/(a+2008)

等式两边相加得

1/a(a+1)+1/(a+1)(a+2)+1/(a+2)(a+3).+1(a+2007)(a+2008)

=1/a-1/(a+1)+1/(a+1)-1/(a+2)+1/(a+2)-1/(a+3).+1/(a+2007)-1/(a+2008)

=1/a+1/(a+2008)

=2008/a(a+2008)