作业帮1/1*3+1/3*5+1/5*7+.+1/49*51怎么算啊?
来源：学生作业帮助网 编辑：作业帮 时间：2024/11/16 05:25:12
1/1*3+1/3*5+1/5*7+.+1/49*51怎么算啊?
1/1*3+1/3*5+1/5*7+.+1/49*51怎么算啊?
1/1*3+1/3*5+1/5*7+.+1/49*51怎么算啊?
1/1×3+1/3×5＋1/5×7+…＋1/49×51
=1/2×[1-1/3]+1/2×[1/3-1/5]+1/2×[1/5-1/7]+...+1/2×[1/49-1/51]
=1/2×[1-1/3+1/3-1/5+1/5-1/7+.+1/49-1/51]
=1/2×[1-1/51]
=1/2×50/51
=25/51
1/1*3就是 (1-1/3)/2
1/3*5就是 (1/3-1/5)/2
你逐项拆开就能发现很多个分数都能消去。
最后剩下(1-1/51)/2
1/1*3=(1/2)×(1/1-1/3)
1/3*5=(1/2)×(1/3-1/5)
1/5*7=(1/2)×(1/5-1/7)
...
1/49*51=(1/2)×(1/49-1/51)
1/1*3+1/3*5+1/5*7+......+1/49*51
=(1/2)×(1/1-1/3+1/3-1/5+1/5-1/7+....+1/49-1/51)
=(1/2)×(1/1-1/51)
=(1/2)×50/51
=25/51
原式等于1/2(1-1/3+1/3-1/5......+1/49-1/51)
=1/2(1-1/51)
=25/51
1/2[2/(1*3)+2/(3*5)+2/(5*7)…2/(49*51)]
=1/2(1-1/3+1/3-1/5+1/5-1/7+…1/49-1/50)
=1/2*50/51
=50/102=25/51
好了
解析：
1/[n(n+2)]=(1/2)[(n+2)-n]/[n(n+2)]=(1/2)[1/n-1/(n+2)]
∴原式
=(1/2)*[1-1/3+1/3-1/5+1/5-1/7+…………+1/49-1/51]
=(1/2)*[1-1/51]
=(1/2)*50/51
=25/51
不懂欢迎再问
谢谢