Question

3分之一+15分之1+35分之1.。。。。。一直加到到99乘以101分之一等于多少？

Answer 1

1/3+1/15+1/35+…+1/(99×101)
=1/(1×3)+1/(3×5)+1/(5×7)+…+1/(99×101)
该式可通过裂项相消法进行求解，需要用到的公式为1/[a(a+2)]=1/2×{(a+2-a)/[a(a+2)]}=1/2×{(a+2)/[a(a+2)]-a/[a(a+2)]}=1/2×[1/a-1/(a+2)]
因此，原式
=1/2×(1/1-1/3)+1/2×(1/3-1/5)+1/2×(1/5-1/7)+…+1/2×(1/99-1/101)
=1/2×(1/1-1/3+1/3-1/5+1/5-1/7+…+1/99-1/101)
=1/2×(1/1-1/101)
=1/2×100/101
=50/101

Answer 2

1/3+1/15+……+1/99*101
=1/3+(1/3-1/5)/2+……+(1/99-1/101)/2
=1/3+(1/3-1/101)/2
=1/3+1/6-1/202
=1/2-1/202
=101/202-1/202
=100/202=50/101