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A linear function in mathematics is expressed in the form \(f(x) = ax + b\), where \(a\) and \(b\) are constants, and \(x\) represents the variable. The function creates a line when it is graphed, hence the name linear. The constant \(a\) is known as the slope, which determines the steepness or direction of the line.

The sequence given in the exercise, \(t_n = 2n - 1\), represents a linear function where the variable \(n\) is the term number. In this sequence, the coefficient \(2\) serves as the slope. It indicates how much the value of \(t_n\) increases as \(n\) increases by one. Specifically, here the slope indicates the sequence increases by 2 for every increase in the term number.

This characteristic of linearity guarantees that the graph of the sequence forms a straight line when plotted, signifying a consistent increment pattern across terms.

Sequence analysis involves examining a sequence's properties to understand its behavior. The provided sequence \(t_n = 2n - 1\) is explicitly defined, meaning each term is directly obtained by plugging \(n\) into the sequence's formula.

By computing terms like \(t_1\), \(t_2\), and \(t_3\), which are respectively 1, 3, and 5, you can observe that the sequence appears to form an arithmetic progression. It's beneficial to check for a common difference, which is the consistent interval between consecutive terms:

• The difference between \(t_2\) and \(t_1\) is 2.
• The difference between \(t_3\) and \(t_2\) is also 2.

These calculations demonstrate that the sequence follows a predictable pattern. Recognizing these patterns is crucial in sequence analysis as it allows predictions about future terms and general sequence behavior.

In the context of the given exercise, inequality comparison helps to determine whether the sequence is increasing or decreasing by comparing consecutive terms. To check if a sequence is decreasing, each term should be greater than the following term, leading to an inequality like \(t_n > t_{n+1}\).

For the sequence \(t_n = 2n - 1\), we determine \(t_{n+1}\) by substituting \(n + 1\) into the expression, giving \(t_{n+1} = 2n + 1\).

Using these expressions, the inequality comparison is set up as \(2n - 1 > 2n + 1\). Simplifying this inequality results in \(-1 > 1\), which is an invalid statement, proving that the sequence is not decreasing.

This systematic process of inequality comparison effectively reveals that the sequence is not diminishing but, in fact, increasing as \(n\) grows.