None of the terms in the inductive sequence a_i are 0

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None of the terms in the inductive sequence a_i are 0

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Source: Baltic Way 1996 Q7

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WakeUp

#1 h Mar 19, 2011, 1:20 PM • 2 Y

Y by Adventure10, Mango247

A sequence of integers $a_1,a_2,\ldots$ is such that $a_1=1,a_2=2$ and for $n\ge 1$ ,
$a_{n+2}=\left\{\begin{array}{cl}5a_{n+1}-3a_{n}, &\text{if}\ a_n\cdot a_{n+1}\ \text{is even},\\ a_{n+1}-a_{n}, &\text{if}\ a_n\cdot a_{n+1}\ \text{is odd},\end{array}\right.$
Prove that $a_n\not= 0$ for all $n$ .

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pco

#2 h Mar 19, 2011, 1:34 PM • 2 Y

Y by Adventure10, Mango247

WakeUp wrote:

A sequence of integers $a_1,a_2,\ldots$ is such that $a_1=1,a_2=2$ and for $n\ge 1$ ,
$a_{n+2}=\left\{\begin{array}{cl}5a_{n+1}-3a_{n}, &\text{if}\ a_n\cdot a_{n+1}\ \text{is even},\\ a_{n+1}-a_{n}, &\text{if}\ a_n\cdot a_{n+1}\ \text{is odd},\end{array}\right.$
Prove that $a_n\not= 0$ for all $n$ .

Looking at the sequence mod $4$ , we get $\{1,2,3,1,2,3,1,2,3,...\}$
Hence the result.

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