Geometric Progression

Let \(\displaystyle x=m+n, \text{ where }m\text{ is an integer, and }0

Then, $\displaystyle \lfloor x \rfloor =m,\text{ and } x- \lfloor x \rfloor=n.$

If the common ratio of the GP is \(\displaystyle r,(0

$\displaystyle (m+n)r=m,\text{ and } mr=n.$

Substitute the second equation into the first and we have, $\displaystyle n(1+r)=m.$

 $\displaystyle r<1, \text{ so }(1+r)<2,$ 

and since n is also less than 1,

it follows that the only possible (positive) integer value for m is m = 1.

That implies that r = n, and substitution into the equation $\displaystyle (m+n)r=1,\text{ gets us }n^{2}+n-1=0,$

from which

$\displaystyle n=\left(-1+\sqrt{5}\right)/2\approx0.618,$

(the golden ratio).

$\displaystyle x\approx1.618.$

Tiggsy