Give three number such that x,[x] and {x} are in geometricprogression?

Question

Give three number such that x,[x] and {x} are in geometricprogression?

Answer 1

wikipedia

Answer 2

I suppose [x] means the integer part of x (the greatest integer, less than or equal to x) and {x} means the fractional part, i.e.

{x} = x - [x] /sometimes denoted int(x) and frac(x)/.

Then the answer is x = (1 + √5)/2 - the  famous Golden Ratio (follow the link in Sources below to read an article), we have

[x] = 1, {x} = (-1 + √5)/2 and x/[x] = [x]/{x}

Indeed let i = [x] = int(x) and f = {x} = frac(x), the progression

i + f, i, f implies (i + f)f = i² or i(i - f) = f², but since

0 ≤ f < 1 the right hand side above f² < 1, hence i = 1 and

(1+ f)f = 1 yields finally f = (-1 + √5)/2 and x = (1 + √5)/2

Answer 3

2,6,18