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

($\lfloor x\rfloor$ means the greatest integer less than or equal to $x$.)

Question

Which positive real number $x$ has the property that $x$, $\lfloor x \rfloor$, and $x - \lfloor x\rfloor$ form a geometric progression (in that order)?

($\lfloor x\rfloor$ means the greatest integer less than or equal to $x$.)

anonymous · Answer

Assume that $ \lfloor{x}\rfloor=1$. Then the common ratios become$$\frac{ 1 }{ x } = x-1$$Multiplying through by $x$ and rearranging gives$$x^2-x-1=0$$Solving this quadratic yields two solutions, only one of which satisfies the question.