Philippine Mathematical Olympiad

Solution:

Let $a_{1}$ be the first term and $r$ the common ratio of the geometric sequence. Since the second, fourth, and fifth terms form an arithmetic sequence,
$$\begin{aligned} a_{1} r^{3} - a_{1} r &= a_{1} r^{4} - a_{1} r^{3} \\ r^{3} - r &= r^{4} - r^{3} \\ 0 &= r^{4} - 2 r^{3} + r \\ 0 &= r\left(r^{3} - 2 r^{2} + 1\right) \\ 0 &= r(r-1)\left(r^{2} - r - 1\right) \\ r &= 0, 1, \frac{1 \pm \sqrt{5}}{2} \end{aligned}$$
Since the geometric sequence has distinct terms and a positive common ratio, the only possible value of the common ratio is $\frac{1+\sqrt{5}}{2}$.