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In mathematics, a sequence is a set of numbers arranged in a particular order following a specific rule, called a sequence formula. For the given problem, the sequence formula is \(a_{n} = \frac{n^{2} - 1}{n^{2} + 1}\). This formula helps you find the terms of the sequence by substituting different values of \(n\). Each value of \(n\) will give you a different term in the sequence. Understanding the rule helps in predicting future terms without calculating each one individually. Analyzing the sequence formula allows us to see how the terms are structured and how they evolve. Using formulas in sequences often makes it easier to work with larger sets of numbers and navigate the patterns they form.

Substitution is a fundamental concept in mathematics, especially when dealing with algebraic expressions and sequences. In this context, substitution means replacing the variable \(n\) in the sequence formula with specific values to find particular terms. For example, to calculate the first term, substitute \(n=1\) in the formula: \(a_{1} = \frac{1^{2} - 1}{1^{2} + 1} = \frac{0}{2} = 0\). For the second term, substitute \(n=2\): \(a_{2} = \frac{2^{2} - 1}{2^{2} + 1} = \frac{3}{5}\). The process of substitution allows us to calculate each required term by plugging in the values 1 through 5. By understanding and applying substitution, solving sequence problems becomes straightforward.

Fractions are a way to represent a part of a whole. In our sequence formula, most terms result in fractions. For example, \(a_{2} = \frac{3}{5}\). A fraction consists of a numerator (top part) and a denominator (bottom part). Simplifying fractions involves finding a common divisor for both the numerator and the denominator. For example, \(a_{3} = \frac{8}{10}\) can be simplified to \(a_{3} = \frac{4}{5}\) by dividing both 8 and 10 by their greatest common divisor, which is 2. Understanding how to work with and simplify fractions is crucial for interpreting the terms in sequences correctly.

Calculating the terms of a sequence involves substituting values into the sequence formula and simplifying the results. Let's go through the steps:

• First Term (\(n=1\)): \(a_{1} = \frac{1^{2} - 1}{1^{2} + 1} = 0\).
• Second Term (\(n=2\)): \(a_{2} = \frac{2^{2} - 1}{2^{2} + 1} = \frac{3}{5}\).
• Third Term (\(n=3\)): \(a_{3} = \frac{3^{2} - 1}{3^{2} + 1} = \frac{8}{10} = \frac{4}{5}\).
• Fourth Term (\(n=4\)): \(a_{4} = \frac{4^{2} - 1}{4^{2} + 1} = \frac{15}{17}\).
• Fifth Term (\(n=5\)): \(a_{5} = \frac{5^{2} - 1}{5^{2} + 1} = \frac{24}{26} = \frac{12}{13}\).

Each term is found by substituting the value of \(n\) into the formula, calculating the numerator and denominator, and simplifying the fraction if needed. This systematic approach ensures accurate calculation of each term.