Question

计算3分之1+15分之1+35分之1+...+9999分之1.

Answer 1

1/3+1/15+1/35+…+1/9999
=1/(1*3)+1/(3*5)+1/(5*7)+…+1/(99*101)
=(1/2)*[1-1/3+1/3-1/5…+1/99-1/101]
=(1/2)*(1-1/101)
=(1/2)*(100/101)
=50/101
用到裂项：1/(ab)=1/|a-b| * |1/a-1/b|
有不懂欢迎追问

Answer 2

分析，
1/15=(1/3-1/5)×1/2
1/35=(1/5-1/7)×1/2
……
1/9999=(1/99-1/101)×1/2

∴1/3+1/15+1/35+……+1/9999
=1/3+(1/3-1/5)×1/2+(1/5-1/7)×1/2+……+(1/99-1/101)×1/2
=1/3+1/2×(1/3-1/5+1/5-1/7+……+1/99-1/101)
=1/3+1/2×(1/3-1/101)
=1/3+49/303
=150/303
=50/101

Answer 3

∴1/3+1/15+1/35+……+1/9999
=1/3+(1/3-1/5)×1/2+(1/5-1/7)×1/2+……+(1/99-1/101)×1/2
=1/3+1/2×(1/3-1/5+1/5-1/7+……+1/99-1/101)
=1/3+1/2×(1/3-1/101)
=1/3+49/303
=150/303
=50/101