14.45 For a positive number like 3.27, 3 is called the integer part of this number, and 0.27 is called the decimal part of this number. Find a positive number such that its decimal part, its integer part, and the number itself are consecutive terms of a geometric sequence.
(7th Canadian Mathematics Competition, 1.975 year)
[Solution] From the given, we have
$$
\frac{[x]}{\{x\}}=\frac{x}{[x]} .
$$

Since $x=[x]+\{x\}$, then
$$
\frac{[x]}{\{x\}}=\frac{[x]+\{x\}}{[x]}=1+\frac{\{x\}}{[x]} .
$$

Since $\{x\},[x], x$ are three terms of a geometric sequence, then
$$
[x] \neq 0 \text {. }
$$

Thus,
$$
[x] \geqslant 1 \text {. }
$$

Therefore, we have
$$
[x]0$ gives $r=\frac{-1+\sqrt{5}}{2}$.
The positive number $x$ is
$$
x=[x]+r=1+\frac{-1+\sqrt{5}}{2}=\frac{1+\sqrt{5}}{2} .
$$