Which positive real number x has the property that x, floor of x and x - floor of x form a geometric progression

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)?

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

Thanks in advance! Don’t worry about the solution, I just need to put the answer into a program so that I can read the solution

I tried plugging in random numbers but that didn’t work

Answer 1

HINT

Geometric progression is a sequence where every pair of consecutive terms has the same ratio. Hence, in your case, if $x = n+a$ where $a\in[0,1)$ and $n \in \mathbb{N}$, then your sequence looks like $n+a,n,a$ and that it's geometric implies $$ \frac{n+a}{n} = \frac{n}{a}. $$ Can you finish this?

Answer 2

Hint:

What the question is saying is that:

$$x=a$$ $$\lfloor x \rfloor = ar$$ $$x-\lfloor x \rfloor=ar^2$$