1981 IMO 第 4 题

Let $\left\{f_{n}\right\}$ be the Fibonacci sequence $\{1,1,2,3,5, \ldots\}$. (a) Find all pairs $(a, b)$ of real numbers such that for each $n, a f_{n}+b f_{n+1}$ is a member of the sequence. (b) Find all pairs $(u, v)$ of positive real numbers such that for each $n$, $u f_{n}^{2}+v f_{n+1}^{2}$ is a member of the sequence.

令 $\left\{f_{n}\right\}$ 为斐波那契数列 $\{1,1,2,3,5, \ldots\}$。 (a) 找到所有实数对 $(a, b)$，使得对于每个 $n，f_{n}+b f_{n+1}$ 是序列的成员。 (b) 找到所有正实数对 $(u, v)$，使得对于每个 $n$，$u f_{n}^{2}+v f_{n+1}^{2}$ 是序列的成员。