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Possible Duplicate:
Recurrence relation, Fibonacci numbers

$(a)$ Consider the recurrence relation $a_{n+2}a_n = a^2 _{n+1} + 2$ with $a_1 = a_2 = 1$.

$(i)$ Assume that all $a_n$ are integers. Prove that they are all odd and the integers $a_n$ and $a_{n+1}$ are coprime for $n \in \mathbb N$

$(ii)$ Assume that the set $\{a_n , a_{n+1} , a_{n+2}\}$ is pairwise coprime for $n \in \mathbb N$. Prove that all $a_n$ are integers by induction.

$(b)$ Consider the recurrence relation $a_{n+2}a_n = a^2_{n+1} + 1$ with $a_1 = 1, a_2 = 2$ and compare this sequence to the Fibonacci numbers. What do you find? Formulate it as a mathematical statement and prove it.

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    You should really do some initial work! Because for example statement (i) is easy to show: by induction show they are odd, then use a simple divisibility argument do finish via induction that two adjacent terms are coprime.2012-11-19
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    http://math.stackexchange.com/questions/236578/recurrence-relation-fibonacci-numbers/2366342012-11-19
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    http://math.stackexchange.com/questions/240712/recurrence-relations-question2012-11-19

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