1除以3×5+1除以5×7+1除以7×9+1除以9×11+.........1除以99×101

1÷(3×5)
=(1/2)×(5-3)÷(3×5)
=(1/2)×[5÷(3×5)-3÷(3×5)]
=(1/2)×(1/3-1/5)

同理
1÷(5×7)
=(1/2)×(7-5)÷(5×7)
=(1/2)×[7÷(5×7)-5÷(5×7)]
=(1/2)×(1/5-1/7)

后面以此类推
所以原式=(1/2)×(1-3-1/5+1/5-1/7+……+1/99-1/101)
=(1/2)×(1/3-1/101)
=49/303

1/n(n+2)=1/2[1/n-1/(n+2)]
1/3*5+1/5*7+1/7*9+……+1/99*101
=1/2(1/3-1/5+1/5-1/7+……+1/99-1/101)
=1/2(1/3-1/101)
=1/2*98/303
=49/303

将每一项都分别拆开
例如
1除以3×5=(1/3-1/5)/2

1除以3×5+1除以5×7+1除以7×9+1除以9×11+.........1除以99×101

=1/2×[(1/3-1/5)+(1/5-1/7)+(1/7-1/9)+(1/9-1/11)+...+(1/99-1/101)]
=1/2×(1/3-1/101)
=49/303

1除以3×5+1除以5×7+1除以7×9+1除以9×11+.........1除以99×101
=1/2((1/3-1/5)+(1/5-1/7)+(1/7-1/9)+(1/9-1/11)+......+(1/99-1/101))
=1/2(1/3-1/101)
=49/303