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阅读材料:因为:1/(1·3)=1/2(1-1/3),1/(3·5)=1/2(1/3-1/5),1/(5·7)=1/2(

来源:学生作业帮 编辑:作业帮 分类:数学作业 时间:2022/08/14 15:16:49
阅读材料:
因为:1/(1·3)=1/2(1-1/3),1/(3·5)=1/2(1/3-1/5),1/(5·7)=1/2(1/5-1/7),……
1/(17·19)=1/2(1/17-1/19)
所以:1/(1·3)+1/(3·5)+1/(5·7)+……+1/(17·19)
=1/2(1-1/3)+1/2(1/3-1/5)+1/2(1/5-1/7)+……+1/2(1/17-1/19)
=1/2(1-1/3+1/3-1/5+1/5-1/7+……+1/17+1/19)
=1/2(1-1/19)
=9/19
受此启发,请解下面的方程:
1/x(x+3)+1/(x+3)(x+6)+1/(x+6)(x+9)=3/(2x+18)
1/x(x+3)=1/3 * [1/x - 1/(x+3)],
1/(x+3)(x+6)=1/3 * [1/(x+3) - 1/(x+6)],
1/(x+6)(x+9)=1/3 * [1/(x+6) - 1/(x+9)],
所以左式
1/x(x+3)+1/(x+3)(x+6)+1/(x+6)(x+9)
= 1/3 * [1/x - 1/(x+9)],
= 1/3 * (x+9-x) / x(x+9)
= 1/3 * 9/ x(x+9)
= 3/ x(x+9)
因为原式左右两边相等,则:
3/x(x+9) = 3/(2x+18)
==>x(x+9) = 2x + 18
==>x^2 + 7x - 18 = 0
==>x1 = 2,x2 = -9 .
因为x= -9代入右式得分母为0,所以舍去.
以 x= 2 代入原式检查:左边= 1/10 + 1/40 + 1/88 = 3/22
右边= 3/(4+18)= 3/22 ,左右两式相等.