When diving into the world of mathematics and computer science, we often encounter sequences of numbers arranged in a particular order. Two terms frequently used in this context are “non-decreasing order” and “increasing order.” Although they might sound similar at first glance, they hold subtly different meanings that can significantly impact the interpretation of data, algorithms, and mathematical proofs. Let’s demystify these terms to understand their distinct uses and implications.

#### Increasing Order: A Strict Progression

A sequence of numbers is said to be in **increasing order** when each number is strictly greater than the one preceding it. This means that no two consecutive numbers in the sequence can be the same; each must be larger than the previous. The emphasis here is on the word “strictly,” which indicates that the progression must always ascend without repetition.

For example, the sequence 3, 7, 11, 18 is in increasing order. Each number moves up without any plateaus. However, the moment we encounter a sequence like 3, 7, 7, 18, the criteria for increasing order are violated because the sequence does not strictly ascend (due to the repetition of 7).

#### Non-Decreasing Order: A Gentle Climb

In contrast, a sequence is in **non-decreasing order** if each number is greater than or equal to the one before it. This definition relaxes the strictness found in increasing order by allowing consecutive numbers to be equal. The critical point here is the allowance for equality, ensuring that the sequence does not descend but can maintain a steady level before possibly ascending again.

A perfect illustration of a non-decreasing sequence is 5, 8, 8, 13, where the sequence does not decrease at any point, although it plateaus at 8. This flexibility makes the term “non-decreasing” particularly useful in mathematical and computational contexts where such plateaus are significant and permissible.