## zmudz one year ago 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 $$x$$.)

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.