${{a}_{n+1}}=\frac{{{a}_{n}}+{{a}_{n+2}}}{2}$ for all $n \ge 1$. Then$\{{{a}_{n}}\}$is unbounded.

Question

Let $\{{{a}_{n}}\}$ be a non-constant sequence in $\mathbb R $ such that ${{a}_{n+1}}=\frac{{{a}_{n}}+{{a}_{n+2}}}{2}$ for all $n \ge 1$. Then $\{{{a}_{n}}\}$ is unbounded.

Geometrically I plotted the sequences and understand inter distance between any two consecutive terms is fixed and $a_2-a_1$. So it is either increasing or decreasing. But how to prove it is unbounded.

Answer 1

Hint

The distance between $a_n-a_1$ is a multiple of $a_2-a_1$. Now tend $n$ to $\infty$.

Answer 2

Notice that $$a_{n+2} = 2a_{n+1} - a_n \iff a_{n+2} - a_{n+1} = a_{n+1}-a_n.$$ Define the sequence $(s_n)$ by $s_n = a_{n+1} - a_n$. Then, $(s_n)$ is a constant sequence, with $s_1 = a_2 - a_1 =: b$. Now, $$ nb = \sum_{k = 1}^n s_k = a_{n+1} - a_1.$$ Therefore, $(a_n)$ is unbounded iff $a_1\neq a_2$.