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
