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In mathematics, a sequence is an ordered list of numbers that follow a particular rule. When evaluating a sequence, you analyze these numbers using the rule to find patterns or predict future values. Understanding sequence evaluation is crucial as it helps in identifying the behavior of complex functions in mathematics and computer science.
In the given exercise, you are required to evaluate sequences like:\[ x_n = 3^n, 2^n, 2^{-n}, \text{ and } 3(-2)^n. \]
You substitute these into the expression \(2 x_{n+2} - 7 x_{n+1} + 3 x_n\). For each sequence option given, such as \(x_n = 3^n\), you calculate subsequent terms \(x_{n+1} = 3^{n+1}\) and \(x_{n+2} = 3^{n+2}\), and plug them into the expression. This step illustrates sequence evaluation by simplification and substitution. Using basic arithmetic rules, you process the expressions to see if they resolve to zero, as required by the recurrence relation\( 2 x_{n+2} - 7 x_{n+1} + 3 x_n = 0 \), which defines the valid sequences.

A homogeneous recurrence relation is a type of recurrence relation where each term is a linear combination of earlier terms, without a constant term added. The relation we have, \( 2 x_{n+2} - 7 x_{n+1} + 3 x_n = 0 \), is considered homogeneous since the right side equals zero.
This indicates that all terms are dependent on each other, and hence, if you find a solution sequence, it shouldn’t require adding any independent constants. Solving involves manipulating terms so that they balance out to maintain the zero on the right.
Homogeneous equations are pivotal because solutions are derived strictly based on previous terms alone, providing a systematic way to approach these specific types of problems. As shown in the exercise, solutions like \( x_n = 3^n \) satisfy the relation due to their inherent property of cancelling each term, fulfilling the equation's demand for homogeneity.

Difference equations express relationships between the terms in a sequence by showing how terms differ from one another. In recurrence relations, they are crucial as they describe how a sequence progresses.
The given recurrence relation \( 2 x_{n+2} - 7 x_{n+1} + 3 x_n = 0 \) is a second-order linear difference equation because it relates terms at positions \( n, n+1, \text{ and } n+2 \). This type of equation helps us form sequences where each term depends on the prior two terms.
Understanding difference equations allows you to work through the steps of the exercise, deciding which of the sequences like \( x_n = 2^n \) satisfies the equation by checking if substituting terms results in a consistent relationship that resolves to zero. They provide the framework for how sequences transform over time based on their mathematical dependency defined in the equation. Recognizing these patterns is essential for solving recurrence relations efficiently.