# Given $a_1 = 1$, $a_2 = 2$, and $a_n = \frac{1}{2}(a_{n-1} + a_{n-2})$ for $n>2$, prove bounded by 1 and 2 [closed]

I attempted using induction to prove it formally but was not sure how to proceed.

Additionally, how can you prove the sequence is neither monotonically increasing nor decreasing?

## closed as off-topic by Martin R, Aqua, Claude Leibovici, Jack D'AurizioOct 18 '17 at 8:39

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• what did you try? one typical approach when proving something hold for all $n$ could be induction :) – mortal Oct 18 '17 at 8:35
• – Martin R Oct 18 '17 at 8:49

For all $n\geq3$ we have $$a_n=\frac{a_{n-1}+a_{n-2}}{2}\geq\frac{1+1}{2}=1.$$ Also, for all $n\geq3$ we have: $$a_n=\frac{a_{n-1}+a_{n-2}}{2}\leq\frac{2+2}{2}=2.$$ Done!