已知x+1/y=1,y+1/z=1,求证z+1/x=1.

Question

已知x+1/y=1,y+1/z=1,求证z+1/x=1.

Answer 1

证：
由x+1/y=1
得x=1-1/y=(y-1)/y
由y+1/z=1
得1/z=1-y
z=1/(1-y)
所以z+1/x
=1/(1-y)+y/(y-1)
=1/(1-y)-y/(1-y)
=(1-y)/(1-y)
=1
证毕