Why can't this sequence be periodic

Question

Let the sequence $\{a_{n}\}$ be such that $$a_{1}=1,a_{2n}=a_{n},a_{4n-1}=0,a_{4n+1}=1,\forall n\in N^{+}.$$

Show that this sequence can't be periodic.

Arguing by contradiction, we assume that there exists a positive integer $T$ such $a_{n+T}=a_{n},\forall n\in N^{+}$. But how to find a contradiction?

score 13 · Accepted Answer · answered Aug 08 '15 at 14:57

13

Sticking with proof by contradiction: suppose $T$ is the minimal period. Then T must be odd (if $T=2t$ then $a_n=a_{2n}=a_{2n+T} =a_{2n+2t}=a_{n+t}$ so $t$ would be a smaller period).

Then $2T = 2\;\; mod(4)$. But then $a_{3+2T}≠a_3$.

answered Aug 08 '15 at 14:57

lulu

70,402

Thank you very simple solution+1 – Aug 08 '15 at 15:05
@KelantanKota please don't forget to accept the answers which solved your question. – Ruslan Aug 08 '15 at 20:49

Why can't this sequence be periodic

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