Question

a=1,b=3,求1/ab+1/(a+2)(b+2)+1/(a+4)(b+4)+.......1/(a+100)(b+100)的值。

Answer 1

若a=1，b=3，则：
1/ab+1/(a+2)(b+2)+1/(a+4)(b+4)+.......1/(a+100)(b+100)
＝1/(1*3) +1/(3*5) +1/(5*7) +...+1/(99*101) +1/(101*103)
=(1/2)*[(1-1/3)+(1/3 -1/5)+(1/5 -1/7)+...+(1/99 -1/101)+(1/101 -1/103)]
=(1/2)*(1-1/103)
=51/103

Answer 2

1/ab+1/(a+2)(b+2)+1/(a+4)(b+4)+.......1/(a+100)(b+100）
=1/(1×3)+1/(3×5)+1/(5×7)+.....+1/(101×103)
=1/2×(1-1/3)+1/2×(1/3-1/5)+1/2×(1/5-1/7)......+1/2×(1/101-1/103)
=1/2×(1-1/3+1/3-1/5+1/5-1/7+......+1/101-1/103)
=1/2×(1-1/103)
=1/2×102/103
=51/103