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

Question

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?

Answer 1

After checking of the base and after assuming of the induction we obtain the following.

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!