7. (Average/Difficult) A geometric sequence has a nonzero first term, distinct terms, and a positive common ratio. If the second, fourth, and fifth terms form an arithmetic sequence, find the common ratio of the geometric sequence.
Answer: $\frac{1+\sqrt{5}}{2}$
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 $\frac{1+\sqrt{5}}{2}$.