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Omni-MATH / Let {fn} be the Fibonacci sequence {1,1,2,3,5,….}.

Problem

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problem

Let {fn}\{fn\} be the Fibonacci sequence {1,1,2,3,5,….}.\{1, 1, 2, 3, 5, \dots.\}. (a) Find all pairs (a,b)(a, b) of real numbers such that for each nn, afn+bfn+1af_n +bf_{n+1} is a member of the sequence. (b) Find all pairs (u,v)(u, v) of positive real numbers such that for each nn, ufn2+vfn+12uf_n^2 +vf_{n+1}^2 is a member of the sequence.
Plain-text mathematical notation (without MathML)
Let {fn} be the Fibonacci sequence {1,1,2,3,5,….}.

(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.
Original LaTeX notation
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.

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Source and history

task_id
9a4b3a5f-0bb6-5836-a400-8ef75023e5b6
task_key
test--9a4b3a5f-0bb6-5836-a400-8ef75023e5b6
task_revision_id
1
config / split
/ test

Official source

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