Can a Sequence Be Both Increasing and Decreasing?

The question of whether a sequence can be both increasing and decreasing simultaneously is a fundamental concept in mathematics. Generally, the answer is a definitive no. However, there are nuanced cases, particularly when considering sequences in a broader sense, that can lead to some interesting scenarios.

Strictly Increasing and Strictly Decreasing Sequences

First, let's clarify the definitions:

Strictly Increasing Sequence: A sequence where each term is greater than the previous term. For example, $a_1 a_2 a_3$. Strictly Decreasing Sequence: A sequence where each term is less than the previous term. For example, $a_1 a_2 a_3$.

By definition, a sequence cannot be both strictly increasing and strictly decreasing at the same time. If a term is greater than the previous term, it cannot simultaneously be less than the previous term.

Non-Strictly Increasing and Decreasing Sequences

However, sequences can be non-strictly increasing or non-strictly decreasing, where terms can be equal or increase/decrease:

Non-Strictly Increasing Sequence: Terms can be equal or increase. For example, $a_1 a_2 a_3$. Non-Strictly Decreasing Sequence: Terms can be equal or decrease. For example, $a_1 a_2 a_3$.

When a sequence is non-strictly increasing and non-strictly decreasing, it can be constant, meaning all its terms are equal. For instance, $a_1 a_2 a_3$.

Probability Distributions and Sequences

In more advanced contexts, such as probability distributions, a sequence can exhibit both increasing and decreasing behavior:

For instance, consider the Poisson probability distribution with probability function $P(X x) e^{-lambda} frac{lambda^x}{x!}$. The sequence of probabilities $P(X 1), P(X 2), P(X 3), dots$ forms a sequence that increases until $x lambda$ and then decreases from that point onward. This is an example where a sequence can transition from increasing to decreasing behavior.

Mixed Increasing and Decreasing Sequences

Some sequences may increase in one interval and decrease in another. For example, consider the sequence defined as:

$a_n sin^2 left( frac{n pi}{6} right)$

Selecting a few values, we get:

$a_1 0$ $a_2 frac{1}{4}$ $a_3 frac{3}{4}$ $a_4 1$ $a_5 frac{3}{4}$ $a_6 frac{1}{4}$ $a_7 0$ $a_8 frac{1}{4}$

This sequence shows a mixed pattern of increases and decreases, which can be observed in certain mathematical functions and distributions.

Conclusion

In summary, while a sequence cannot be both strictly increasing and strictly decreasing simultaneously, it can be both increasing and decreasing in a broader sense, particularly when considering non-strictly monotonic sequences or special cases such as probability distributions and mixed behavior sequences.