Problem 29 - College Algebra with Modeling and Visualization

Question

The first five terms of an arithmetic sequence are given. Find (a) numerical, (b) graphical, and (c) symbolic representations of the sequence. Include at least eight rems of the sequence for the graphical and numerical representations. $$ 1,3,5,7,9 $$

Step-by-Step Solution

Answer

Numerical: 1, 3, 5, 7, 9, 11, 13, 15. Graphical: plot points (1,1), (2,3), ..., (8,15). Symbolic: \(a_n = 2n - 1\).

Step 1: Identify Pattern

Examine the given sequence: 1, 3, 5, 7, 9. Notice that each term increases by 2 from the previous term. This constant increment indicates that it is an arithmetic sequence with a common difference, \(d = 2\).

Step 2: Find the First Term

The first term of the sequence is \(a_1 = 1\). This is the initial value of the sequence from which each term begins to increase by the common difference.

Step 3: Write the Formula for the nth Term

The formula for the nth term of an arithmetic sequence is \(a_n = a_1 + (n-1) imes d\). Here, \(a_1 = 1\) and \(d = 2\), so the formula becomes \(a_n = 1 + (n-1) imes 2 = 2n - 1\).

Step 4: List the First Eight Terms for Numerical Representation

Using the formula \(a_n = 2n - 1\), calculate the first eight terms: 1, 3, 5, 7, 9, 11, 13, 15. This is the numerical representation of the sequence.

Step 5: Graph the Sequence for Graphical Representation

Plot the sequence \((n, a_n)\) for \(n = 1\) to \(n = 8\). The points will be (1,1), (2,3), (3,5), (4,7), (5,9), (6,11), (7,13), (8,15). You'll find that these points lie on a straight line, indicating a linear relationship.

Step 6: Provide Symbolic Representation

The symbolic representation of the sequence is simply the rule or formula that defines it, \(a_n = 2n - 1\), describing how to calculate any term in the sequence.

Key Concepts

Numerical Representation

In an arithmetic sequence like the one given, each term which follows grows by a constant difference from the term before it. For this sequence, the difference, or step, between consecutive terms is 2. This consistent difference is what defines it as arithmetic. To see the complete pattern clearly, we calculate eight terms, letting us see the growth gap further. Starting with the first term, which is 1, we continue by adding 2 successively:

• First term: 1
• Second term: 3 (1 + 2)
• Third term: 5 (3 + 2)
• Fourth term: 7 (5 + 2)
• Fifth term: 9 (7 + 2)
• Sixth term: 11 (9 + 2)
• Seventh term: 13 (11 + 2)
• Eighth term: 15 (13 + 2)

These eight terms (1, 3, 5, 7, 9, 11, 13, 15) numerically show how the sequence advances by 2 each step.

Graphical Representation

To provide a visual understanding of the arithmetic sequence, we graph the pairs \((n, a_n)\), where \(a_n\) shows the sequence term corresponding to \(n\). For the first eight terms calculated previously, our pairs are:

• (1, 1)
• (2, 3)
• (3, 5)
• (4, 7)
• (5, 9)
• (6, 11)
• (7, 13)
• (8, 15)

When these points are plotted on a graph, they lie on a straight line because of the arithmetic nature of the sequence. The linear relationship shows the steady increase of 2 per step and visually confirms the structure of an arithmetic sequence.

Symbolic Representation

The symbolic representation of an arithmetic sequence uses a neat and simple formula that describes how any term in the sequence is determined. For our given sequence, we use the equation \(a_n = a_1 + (n-1) \times d\). Using the first term \(a_1 = 1\) and the common difference \(d = 2\), the formula becomes \(a_n = 2n - 1\). This formula tells us how to find any term throughout the sequence. Simply plug any positive integer in for \(n\), and you'll have the term for that position. Thus, the symbolic form is a concise way to encapsulate the whole sequence in a single line.