 Let’s discuss a problem and find out whether a sequence is bounded or unbounded from TIFR 2013 Problem 35. Try before reading the solution.

Question: TIFR 2013 problem 35

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 the 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 is non-zero. Hence the sequence has to be unbounded.