If $y,x,z$ are in Arithmetic Progression(AP) and $z,y,x$ are in Geometric Progression(GP) then prove that: $x,z,y$ are in Harmonic Progression (HP).

Question

If $y,x,z$ are in Arithmetic Progression(AP) and $z,y,x$ are in Geometric Progression(GP) then prove that: $x,z,y$ are in Harmonic Progression (HP).

My Attempt:

$y,x,z$ are in AP so, $$x=\dfrac {y+z}{2}$$ $$2x=y+z$$

$z,y,x$ are in GP then: $$y^2=xz$$ $$2y^2=2xz$$ $$2y^2=(y+z)z$$ $$\dfrac {2y^2}{y+z}=z$$

Answer 1

$\left\{\begin{matrix}2x=y+z\\ y^2=xz\end{matrix}\right.\Rightarrow \left\{\begin{matrix}\frac2z=\frac1x+\frac{y}{xz}\\ y^2=xz\end{matrix}\right.\Rightarrow \frac2z=\frac1x+\frac1{y}$

Answer 2

HINT:

WLOG $y=x-d,z=x+d$

If $d=0$ we are done

Else

$y^2=zx\iff(x-d)^2=(x+d)x\iff d^2=3xd\implies d=3x$