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?

what did you try? one typical approach when proving something hold for all $n$ could be induction :) — JKay, Oct 18 '17 at 08:35
https://math.stackexchange.com/q/1330605, https://math.stackexchange.com/q/1928249, https://math.stackexchange.com/q/2276402, https://math.stackexchange.com/q/502100, https://math.stackexchange.com/q/1451455, https://math.stackexchange.com/q/514083 — Martin R, Oct 18 '17 at 08:49

Michael Rozenberg · Answer 1 · 2017-10-18T08:37:50.183

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!

edited Oct 18 '17 at 08:37

answered Oct 18 '17 at 08:36

Michael Rozenberg

194,933

To argue this formally you would have to use induction, but yeah this is a general idea. – Siddhartha Oct 18 '17 at 08:37

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

1 Answers1