SCHWEITZER 1985 Problem 8

Let $\frac{2}{\sqrt5+1}\leq p < 1$ , and let the real sequence $\{ a_n \}$ have the following property: for every sequence $\{ e_n \}$ of $0$ 's and $\pm 1$ 's for which $\sum_{n=1}^\infty e_np^n=0$ , we also have $\sum_{n=1}^\infty e_na_n=0$ . Prove that there is a number $c$ such that $a_n=cp^n$ for all $n$ . [Z. Daroczy, I. Katai]