Problem 40 - College Algebra with Modeling and Visualization

Question

The first five terms of a geometric sequence are given. Find (a) numerical, (b) graphical, and (c) symbolic representations of the sequence. Include at least eight terms of the sequence for the graphical and numerical representations. $$9,6,4, \frac{8}{3}, \frac{16}{9}$$

Step-by-Step Solution

Answer

Common ratio: \( \frac{2}{3} \). Sequence: 9, 6, 4, \( \frac{8}{3} \), \( \frac{16}{9} \), \( \frac{32}{27} \), \( \frac{64}{81} \), \( \frac{128}{243} \). Symbolic: \( a_n = 9 \times \left( \frac{2}{3} \right)^{n-1} \).

Step 1: Identify the Common Ratio

The common ratio \( r \) in a geometric sequence is the factor by which each term is multiplied to get the next term. To find \( r \), divide the second term by the first term. Here, \( r = \frac{6}{9} = \frac{2}{3} \).

Step 2: Verify the Common Ratio

Verify that the common ratio is consistent by dividing each subsequent term by its preceding term. Ensure that each division yields \( r = \frac{2}{3} \). Since \( \frac{4}{6} = \frac{2}{3} \), \( \frac{8/3}{4} = \frac{2}{3} \), and \( \frac{16/9}{8/3} = \frac{2}{3} \), the common ratio is consistent.

Step 3: Extend the Sequence to Eight Terms

Start with the first term, 9, and multiply by the common ratio repeatedly to find the next terms: 9, 6, 4, \( \frac{8}{3} \), \( \frac{16}{9} \), \( \frac{32}{27} \), \( \frac{64}{81} \), \( \frac{128}{243} \).

Step 4: Present the Numerical Representation

The numerical representation of the sequence is: 9, 6, 4, \( \frac{8}{3} \), \( \frac{16}{9} \), \( \frac{32}{27} \), \( \frac{64}{81} \), \( \frac{128}{243} \).

Step 5: Plot the Graphical Representation

Plot the terms as points on a graph with the term number on the x-axis and the term value on the y-axis. The sequence will show a downward trend as the values decrease with each term.

Step 6: Define the Symbolic Representation

A geometric sequence can be represented by the formula \( a_n = a_1 \times r^{n-1} \). Here, \( a_1 = 9 \) and \( r = \frac{2}{3} \). Thus, the nth term is \( a_n = 9 \times \left( \frac{2}{3} \right)^{n-1} \).

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is a key factor. It helps in understanding how each term relates to the next. The common ratio, denoted as \( r \), is what you multiply one term by to get the subsequent term. This is important because it ensures the sequence follows a consistent pattern.

For the given sequence, \( 9, 6, 4, \frac{8}{3}, \frac{16}{9} \), the common ratio is found by dividing the second term by the first. For this sequence, \( r = \frac{6}{9} = \frac{2}{3} \).

To confirm this ratio across the whole sequence, divide each term by the previous one:

• \( \, \frac{4}{6} = \frac{2}{3} \)
• \( \, \frac{8/3}{4} = \frac{2}{3} \)
• \( \, \frac{16/9}{8/3} = \frac{2}{3} \)

This consistency in the ratio is what makes the sequence geometric.

Numerical Representation

The numerical representation of a geometric sequence is simply listing the terms in a sequential order. This representation helps you see the terms plainly without needing to visualize them further.

To extend a sequence, multiply each term by the common ratio. For our sequence, this means starting with the first term, 9, and applying the common ratio \( \frac{2}{3} \) to get each successive term. The sequence grows as follows:

• 1st term: 9
• 2nd term: 9 \( \times \frac{2}{3} = 6 \)
• 3rd term: 6 \( \times \frac{2}{3} = 4 \)
• 4th term: 4 \( \times \frac{2}{3} = \frac{8}{3} \)
• 5th term: \( \frac{8}{3} \times \frac{2}{3} = \frac{16}{9} \)
• 6th term: \( \frac{16}{9} \times \frac{2}{3} = \frac{32}{27} \)
• 7th term: \( \frac{32}{27} \times \frac{2}{3} = \frac{64}{81} \)
• 8th term: \( \frac{64}{81} \times \frac{2}{3} = \frac{128}{243} \)

The numerical representation provides an easy way to read and understand the sequence's pattern.

Graphical Representation

A graphical representation of a geometric sequence helps visualize how the sequence changes over time. To plot, you'll set the term number on the x-axis and the term value on the y-axis.

As you plot each term:

• The sequence starts at a higher point and moves downward as each term decreases.
• The slope will have a consistent pattern downward because the same common ratio is applied between terms.

Visualizing the sequence graphically makes recognizing patterns and understanding the overall trend much easier.

Symbolic Representation

Symbolic representation is a formal way to express a geometric sequence using algebra. It allows you to find any term in the sequence by using a formula, rather than calculating each step by hand.

For a geometric sequence, the formula for the \( n \)-th term is:\[ a_n = a_1 \times r^{n-1} \]Where:

• \( a_n \) is the \( n \)-th term.
• \( a_1 \) is the first term of the sequence (9 in this case).
• \( r \) is the common ratio \( \left( \frac{2}{3} \right) \).
• \( n \) is the term number.

Thus, the formula for our sequence's terms is \( a_n = 9 \times \left( \frac{2}{3} \right)^{n-1} \). Using this formula, you can quickly calculate any term in the sequence just by inserting the term number \( n \).