Geometric progression of terms involving floor function Which positive real number has the property that $x ,\lfloor x\rfloor,$ and $x-\lfloor x\rfloor$ form a geometric progression (in that order)? 

For $x=\frac{1+\sqrt 5}{2}$, since we have
$$x=\frac{1+\sqrt 5}{2},\ \ \lfloor x\rfloor=1,\ \ x-\lfloor x\rfloor=\frac{-1+\sqrt 5}{2}$$
we have
$$\lfloor x\rfloor=xr,\ \ \ x-\lfloor x\rfloor=xr^2$$
where $r=\frac{-1+\sqrt 5}{2}.$
