1953 AHSME Problem 25

In a geometric progression whose terms are positive, any term is equal to the sum of the next two following terms. then the common ratio is:

\textbf{(A)}\ 1 \qquad \textbf{(B)}\ \text{about }\frac{\sqrt{5}}{2} \qquad \textbf{(C)}\ \frac{\sqrt{5}-1}{2}\qquad \textbf{(D)}\ \frac{1-\sqrt{5}}{2}\qquad \textbf{(E)}\ \frac{2}{\sqrt{5}}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.