Geometric progression of terms involving floor function

Question

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

Answer 1

Hint. Let $ x = a + k $, where $ a = \left\lfloor x \right\rfloor $ and $ 0 \le k < 1 $. Then, we want $$ \frac {k}{a} = \frac {a}{a + k} \implies ak + k^2 = a^2, $$so $ak+k^2$ needs to be an integer.

Answer 2

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 $r=\frac{-1+\sqrt 5}{2}.$