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When working with sequences in algebra, one important task is to evaluate specific terms within the sequence. Each term is represented by a formula often involving the variable \(n\). To find a specific term, you simply substitute the given value of \(n\) into the formula.

For example, if we are given the formula:

\[ a_{n}=\frac{5n-9}{3n+8} \], to find the term \(a_{16}\), we must substitute \(n = 16\). Understanding the evaluation process can help accurately identify any term in a sequence.

This step-by-step evaluation ensures that we handle both the numerator and the denominator correctly. This is crucial, as simple errors could lead to incorrect answers.

Remember: Always check your math when substituting values to avoid mistakes.

The substitution method is straightforward in sequences. This method involves replacing the variable \(n\) with the given value. For the sequence formula:

\[ a_{n}=\frac{5n-9}{3n+8} \], when asked to find \(a_{16}\), replace \(n\) with 16.

So:

\[ a_{16} = \frac{5(16) - 9}{3(16) + 8} \]

Steps to follow:

• Identify the variable to substitute.
• Replace every instance of the variable in the formula with the given value.
• Perform the arithmetic operations as usual.

By substituting correctly, you maintain the sequence's integrity and ensure accurate term evaluation.

Sequences often involve rational expressions. A rational expression is a ratio of two polynomial expressions. For example:

\[ a_{n}=\frac{5n-9}{3n+8} \]

Here, both the numerator \[ 5n - 9 \] and the denominator \[ 3n + 8 \] are polynomials.

When evaluating a specific term, such as \(a_{16}\), you substitute the value into the expression and follow these steps:

• Evaluate the polynomial in the numerator first.


• Evaluate the polynomial in the denominator next.


• Simplify the resulting fraction if possible.

Understanding the structure of rational expressions is fundamental. It allows you to recognize how terms are formed and evaluated in sequences, ensuring greater accuracy in your solutions.