Question:

True/False?

Let $$\{a_n\}$$ be any 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.

Hint:

The given expression is same as $$a_{n+1}-a_n = a_{n+2} -a_{n+1}$$.

Discussion:

The distance between two successive terms in the given sequence is constant. It is given by $$|a_{n+1}- a_n| = |a_n – a_{n-1}| = … = |a_1-a_0|$$.

So for the sequence to be non-constant, $$a_1 \ne a_0$$. Because otherwise the sequence will have distance between any two successive terms zero, which is just another way of saying that the sequence is constant.

There are two cases:

Case 1: $$a_1 > a_0$$.

Then $$a_{n+1} > a_n$$, that is the sequence is increasing, and not only that, it is an arithmetic progression with common difference $$a_1-a_0 (> 0)$$. Therefore, the sequence is unbounded above.

Case 2: $$a_1<a_0$$.

Then as in previous case the sequence this time will become a decreasing sequence, not only that, it is an arithmetic progression with common difference $$< 0$$. Therefore, the sequence is unbounded below.

Remark:

We don’t really need to take the two cases. The key point is that the given recurrence relation is that of an arithmetic progression whose common difference in non-zero. Hence the sequence has to be unbounded.