巧算 (1+1/2+1/3+1/4)(1/2+1/3+1/4+1/5)-(1+1/2+1/3+1/4+1/5)(1/2+1/3+1/4)

来源：学生作业帮助网编辑：作业帮时间：2024/05/08 19:53:40

巧算 (1+1/2+1/3+1/4)*(1/2+1/3+1/4+1/5)-(1+1/2+1/3+1/4+1/5)*(1/2+1/3+1/4)
巧算 (1+1/2+1/3+1/4)*(1/2+1/3+1/4+1/5)-(1+1/2+1/3+1/4+1/5)*(1/2+1/3+1/4)

巧算 (1+1/2+1/3+1/4)*(1/2+1/3+1/4+1/5)-(1+1/2+1/3+1/4+1/5)*(1/2+1/3+1/4)
令a=1:2+1:3+1:4
则1+1:2+1:3+1:4=1+a
1:2+1:3+1:4+1:5=a+1:5
1+1:2+1:3+1:4+1:5=1+a+1:5
所以原式=(1+a)(a+1:5)-a(1+a+1:5)
=a(1+a)+(1+a)×1:5-a(1+a)-a×1:5
=1×1:5+a×1:5-a×1:5
=1:5

设1/2+1/3+1/4=x
1/2+1/3+1/4+1/5=y
(1+1/2+1/3+1/4)*(1/2+1/3+1/4+1/5)-(1+1/2+1/3+1/4+1/5)*(1/2+1/3+1/4)
=(1+x)y-(1+y)x
=y+xy-x-xy
=y-x
=1/5

(1+1/2+1/3+1/4)*(1/2+1/3+1/4+1/5)-(1+1/2+1/3+1/4+1/5)*(1/2+1/3+1/4)
= (1/2+1/3+1/4)*(1/2+1/3+1/4+1/5）+1×(1/2+1/3+1/4+1/5)-1×(1/2+1/3+1/4)-
（1/2+1/3+1/4+1/5)*(1/2+1/3+1/4)
=1×(1/2+1/3+1/4+1/5)-1×(1/2+1/3+1/4)
=1×(1/2+1/3+1/4)+1×1/5-1×(1/2+1/3+1/4)
=1/5

（1+x)*y-(1+y)*x= y-x 所以原式+1/5

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