If x,y and z are in A.P ax,by and cz in G.P and a, b, c in H.P then prove that `x/z+z/x=a/c+c/a`

Explanation

By the given conditions, we have
`2y=x+z (1)`
`b^(2)y^(2)=axxxcz(2)`
`b=2ac/(a+c) (3)`
It is clear from the question that we have to elimanate b and y.
Substituting for y and b from (1) and (3) in (2), we get
`(4a^(2)c^(2))/((a+c)^(2))xx((x+z)^(2))/4=axcz`
or `((x+z)^(2))/(xz)=((a+c)^(2))/(ac)`
or `(x^(2)+z^(2)+2xz)/(xz)=(a^(2)+c^(2)+2ac)/(ac)`
or `x/z+z/x+2=a/c+c/a+2`
or `x/z+z/x=a/c+c/a`