Problem 6 - College Algebra with Modeling and Visualization

Question

Find the first four terms of the sequence. \(a_{n}=5-\frac{1}{n^{2}}\)

Step-by-Step Solution

Answer

The first four terms are 4, \(\frac{19}{4}\), \(\frac{44}{9}\), and \(\frac{79}{16}\).

Step 1: Identify the Formula

The formula for the sequence is given by: \( a_{n} = 5 - \frac{1}{n^2} \). We will use this formula to calculate the terms of the sequence for different values of \(n\).

Step 2: Calculate the First Term

To find the first term, substitute \(n = 1\) into the formula. So, \( a_{1} = 5 - \frac{1}{1^2} = 5 - 1 = 4 \). Therefore, the first term is 4.

Step 3: Calculate the Second Term

For the second term, substitute \(n = 2\) into the formula. Thus, \( a_{2} = 5 - \frac{1}{2^2} = 5 - \frac{1}{4} = \frac{20}{4} - \frac{1}{4} = \frac{19}{4} \approx 4.75 \). The second term is \(\frac{19}{4}\).

Step 4: Calculate the Third Term

Substitute \(n = 3\) into the formula for the third term. Thus, \( a_{3} = 5 - \frac{1}{3^2} = 5 - \frac{1}{9} = \frac{45}{9} - \frac{1}{9} = \frac{44}{9} \approx 4.89 \). The third term is \(\frac{44}{9}\).

Step 5: Calculate the Fourth Term

Finally, substitute \(n = 4\) into the formula. Therefore, \( a_{4} = 5 - \frac{1}{4^2} = 5 - \frac{1}{16} = \frac{80}{16} - \frac{1}{16} = \frac{79}{16} \approx 4.94 \). The fourth term is \(\frac{79}{16}\).

Key Concepts

Sequence Formula

Sequences are a vital part of mathematics and can be found in various forms. A sequence is essentially a list of numbers following a specific pattern. One common way to describe a sequence is through a formula, known as the sequence formula.
This formula provides a rule for calculating any term in the sequence based on its position, usually denoted by the variable \( n \). For example, the sequence formula in this exercise is given by:

• \( a_{n} = 5 - \frac{1}{n^2} \)

This means that to find any particular term, you simply plug in the value of \( n \), which represents the term's position in the sequence.
A clear understanding of the sequence formula allows you to easily predict the behavior of the entire sequence without computing each term individually.

Term Calculation

Term calculation involves finding the specific numbers in a sequence by substituting the values of \( n \) into the sequence formula. This is essential for understanding how sequences develop and change over their progression. Let's break this down:

• First Term: To find the first term, substitute \( n = 1 \) into the formula: \( a_{1} = 5 - \frac{1}{1^2} = 4 \).
• Second Term: For the second term, use \( n = 2 \) in the formula: \( a_{2} = 5 - \frac{1}{2^2} = \frac{19}{4} \).
• Third Term: Substituting \( n = 3 \) gives: \( a_{3} = 5 - \frac{1}{3^2} = \frac{44}{9} \).


• Fourth Term: Finally, \( n = 4 \) results in: \( a_{4} = 5 - \frac{1}{4^2} = \frac{79}{16} \).

Each calculation follows a similar pattern - replacing \( n \) with the term number, applying operations, and simplifying to get the value for that term. This reveals the sequence's structure, showing how it evolves term by term.

Algebraic Sequences

Algebraic sequences are series of numbers where each term is derived using algebraic operations, and they are described using algebraic expressions or formulas. In our exercise, the sequence formula involves subtraction and fraction operations centered around the variable \( n \).
Understanding algebraic sequences requires familiarity with algebraic manipulation, such as simplifying fractions and performing accurate operations. By following the operations step-by-step and substituting different values for \( n \), you can observe how the sequence behaves as \( n \) increases.Some typical characteristics of algebraic sequences include:

• The change between successive terms becomes predictable based on the formula's structure.
• Such sequences can exhibit various patterns, such as linear, quadratic, or more complex relationships.

Algebraic sequences are prevalent in mathematical explorations and real-world applications, providing profound insights into growth patterns, series summations, and mathematical modeling. With practice, the progression of these sequences becomes intuitive and allows for deeper mathematical interpretations.