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An increasing sequence is a sequence where each term is larger than the previous term. This means that as you move from one term to the next, the value either increases or stays the same but does not decrease.
For the sequence given, where each term is defined as \(a_{n} = \frac{n-1}{n+1} \), we can check if it is increasing by comparing each term with the next one.
To determine whether a sequence is increasing, you can use the difference between consecutive terms. If \(a_{n+1} - a_n > 0\), the sequence is increasing. This comparison is an effective way to analyze how a sequence behaves.

Calculus sequences often involve limits, derivatives, or integrals, but in this context, we are primarily focused on sequences as mathematical entities alone. Sequences are ordered lists of numbers, and each number in the list is called a "term."
In calculus, sequences can be approached in a similar manner to functions, analyzing how they change over their "domain." The sequence \(a_n = \frac{n-1}{n+1}\) involves simple arithmetic that can be broken into easy steps for analysis.
This type of sequence can be understood deeply through calculus by observing how their properties affect limits and growth. For instance, in a college calculus class, students might find the limit of a sequence and determine its convergence.

Term comparison is crucial in determining the behavior of sequences. By calculating \(a_{n+1}\) and \(a_n\), and then finding their difference, we can make meaningful comparisons.
In the problem, \(a_{n+1} = \frac{n+1}{n+3}\) and \(a_n = \frac{n-1}{n+1}\) are the consecutive terms we need to compare.
By calculating the difference \(a_{n+1} - a_n\), we determine how the sequence changes. If this difference is positive, it confirms the sequence is increasing; if negative, the sequence decreases. The key is in checking the sign of the difference, simplifying the expression, and understanding its implications.

A positive difference between consecutive terms is the defining characteristic of an increasing sequence. In this exercise, you calculate \(a_{n+1} - a_n = \frac{2}{(n+3)(n+1)}\), a positive value for all \(n \geq 1\).
This tells us that each term is larger than the last, confirming the sequence grows.
The positive difference ensures that for each step forward in the sequence, you observe a positive growth, which parallels the idea of a consistent upward trend. Recognizing a positive difference simplifies understanding the nature of sequences, making determination straightforward.