Let
![\{fn\}](/media/m/5/5/e/55e8ede34194b88474d107d4b3e06589.png)
be the Fibonacci sequence
![\{1, 1, 2, 3, 5, \dots.\}.](/media/m/4/9/6/496fbc2b5add1cf809cb4b474f4db6bc.png)
(a) Find all pairs
![(a, b)](/media/m/0/5/9/059101434c13a86d8297575c7ee676ea.png)
of real numbers such that for each
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
,
![af_n +bf_{n+1}](/media/m/1/1/b/11b3c6d49b2ddc7f0cfd1170880073dc.png)
is a member of the sequence.
(b) Find all pairs
![(u, v)](/media/m/e/6/e/e6ec6f3df818ac4eca8bd19b2176bd61.png)
of positive real numbers such that for each
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
,
![uf_n^2 +vf_{n+1}^2](/media/m/9/c/7/9c74cd3232ed5cca12b57d95f79f7eed.png)
is a member of the sequence.
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Let $\{fn\}$ be the Fibonacci sequence $\{1, 1, 2, 3, 5, \dots.\}.$
(a) Find all pairs $(a, b)$ of real numbers such that for each $n$, $af_n +bf_{n+1}$ is a member of the sequence.
(b) Find all pairs $(u, v)$ of positive real numbers such that for each $n$, $uf_n^2 +vf_{n+1}^2$ is a member of the sequence.