百度作业网1/35+1/63+1/99+1/143 简便计算
来源：学生作业帮 编辑：百度作业网作业帮 分类：数学作业 时间：2024/05/12 18:12:14
1/35+1/63+1/99+1/143 简便计算
1/35+1/63+1/99+1/143
=1/(5×7)+1/(7×9)+1/(9×11)+1/(11×13)
=1/2×(1/5-1/7)+1/2×(1/7-1/9)+1/2×(1/9-1/11)+1/2×(1/11-1/13)
=1/2×(1/5-1/7+1/7-1/9+1/9-1/11+1/11-1/13)
=1/2×(1/5-1/13)
=1/2×(13/65-5/65)
=1/2×8/65
=4/65
再问： 为啥会有1/2
再答： 1/n-1/(n+2)
=(n+2)/[n(n+2)]-n/[n(n+2)]
=2/[n(n+2)]
∴1/[n(n+2)]=1/2×[1/n-1/(n+2)]
=1/(5×7)+1/(7×9)+1/(9×11)+1/(11×13)
=1/2×(1/5-1/7)+1/2×(1/7-1/9)+1/2×(1/9-1/11)+1/2×(1/11-1/13)
=1/2×(1/5-1/7+1/7-1/9+1/9-1/11+1/11-1/13)
=1/2×(1/5-1/13)
=1/2×(13/65-5/65)
=1/2×8/65
=4/65
再问： 为啥会有1/2
再答： 1/n-1/(n+2)
=(n+2)/[n(n+2)]-n/[n(n+2)]
=2/[n(n+2)]
∴1/[n(n+2)]=1/2×[1/n-1/(n+2)]