作业帮1除以(X+1)(X+3)加上1除以(X+3)(X+5)加上1除以(X+5)(X+7)加上1除以（X+7)(X+9)加上1除以（X+9)(X+11)..求上述问题的算法与规律!
来源：学生作业帮助网 编辑：作业帮 时间：2024/04/28 23:05:11
1除以(X+1)(X+3)加上1除以(X+3)(X+5)加上1除以(X+5)(X+7)加上1除以（X+7)(X+9)加上1除以（X+9)(X+11)..
求上述问题的算法与规律!
1÷(x+1)(x+3)+1÷(x+3)(x+5)+1÷(x+5)(x+7)+1÷(x+7)(x+9)+……+1÷(x+2n-1)(x+2n+1)
=1/(x+1)(x+3)+1/(x+3)(x+5)+1/(x+5)(x+7)+1/(x+7)(x+9)+……+1/(x+2n-1)(x+2n+1)
=[1/(x+1)-1/(x+3)]/2+[1/(x+3)-1/(x+5)]/2+[1/(x+5)-1/(x+7)]/2+……+[1/(x+2n-1)-1/(x+2n+1)]/2
=1/[2(x+1)]-1/[2(x+3)]+1/[2(x+3)]-1/[2(x+5)]+1/[2(x+5)]-1/[2(x+7)]+……+1/[2(x+2n-1)]-1/[2(x+2n+1)]
=1/[2(x+1)]-1/[2(x+2n+1)]
=(x+2n+1-x-1)/[2(x+1)(x+2n+1)]
=n/[(x+1)(x+2n+1)]
楼主追问“我不是他舅”先生：一直加到101,结果是?
楼主这里说的101,应该是一直加到1÷/(x+101)(x+103)吧?
套用上面我推导出来的公式,应该是n=51
有：
1÷(x+1)(x+3)+1÷(x+3)(x+5)+1÷(x+5)(x+7)+1÷(x+7)(x+9)+……+1÷(x+101)(x+103)
=51/[(x+1)(x+2×51+1)]
=51/[(x+1)(x+103)]
要是还想继续算的话,就是把分母写成多项式的形式.那么,楼主所求的结果就是：
51/(x^2+104x+103)
0.5*【1/(X+1)-1/(X+3)】+
0.5*【1/(X+3)-1/(X+5)】+
0.5*【1/(X+5)-1/(X+7)】+
0.5*【1/(X+7)-1/(X+9)】+
0.5*【1/(X+9)-1/(X+11)】+······
=0.5*【1/(X+1)-1/(X+3）+1/(X+3)-1/(X+5）+1/(X+5)-1/(X+7）+····...
全部展开
0.5*【1/(X+1)-1/(X+3)】+
0.5*【1/(X+3)-1/(X+5)】+
0.5*【1/(X+5)-1/(X+7)】+
0.5*【1/(X+7)-1/(X+9)】+
0.5*【1/(X+9)-1/(X+11)】+······
=0.5*【1/(X+1)-1/(X+3）+1/(X+3)-1/(X+5）+1/(X+5)-1/(X+7）+······+1/(X+2n-1)-1/(X+2n+1)】
=0.5*【1/(X+1)-1/(X+2n+1)】
=n/[(X+1)(X+2n+1)]
收起
原式=1/2[1/(X+1)-1/(X+3)+1/(X+3)-1/(X+5)+1/(X+5)-1/(X+7)+1/X+7)-1/(X+9)+1/（X+9)-1/(X+11)]
=1/2[1/(x+1)-1/(x+11)]
=5/(X²+12X+11)