Problem 12 - College Algebra with Corequisite Support

Question

For the following exercises, determine whether the sequence is geometric. If so, find the common ratio. \(6,8,11,15,20, \ldots\)

Step-by-Step Solution

Answer

The sequence is not geometric.

Step 1: Identify the nature of the sequence

A geometric sequence is identified by a common ratio between consecutive terms. Therefore, check whether a constant ratio exists between each term of the sequence.

Step 2: Calculate ratios between consecutive terms

Divide each term by the previous term to see if there's a consistent ratio: \( \frac{8}{6} = \frac{4}{3} \)\( \frac{11}{8} = \frac{11}{8} \)\( \frac{15}{11} = \frac{15}{11} \)\( \frac{20}{15} = \frac{4}{3} \)

Step 3: Analyze the calculated ratios

Examine the calculated ratios: - Ratio between first and second term = \( \frac{4}{3} \)- Ratio between second and third term = \( \frac{11}{8} \)- Ratio between third and fourth term = \( \frac{15}{11} \)- Ratio between fourth and fifth term = \( \frac{4}{3} \)The ratios are not the same, indicating the sequence is not geometric.

Key Concepts

Common Ratio

The concept of a common ratio is the defining characteristic of a geometric sequence.
A geometric sequence is one in which each term after the first is obtained by multiplying the previous term by a fixed, constant number called the common ratio. This ratio remains the same, regardless of which terms you choose from the sequence.

To find the common ratio in practice, simply pick any two consecutive terms in a sequence and divide the second term by the first. For example, if you have a sequence with terms 3 and 6, the common ratio is determined by the calculation:

• Common ratio = \( \frac{6}{3} = 2 \)

A consistent common ratio across the sequence indicates that the sequence follows a geometric pattern. If you find that the ratio between terms varies, as seen in the given sequence (e.g., \( \frac{8}{6} \), \( \frac{11}{8} \), etc.), then the sequence is not geometric.

Sequence Analysis

Sequence analysis involves inspecting the structure and progression of a list of numbers.
When dealing with sequences, one of your first tasks is to determine the type of sequence you're examining. Is it arithmetic, geometric, or neither?

For geometric sequences, analyze the relationships between terms by examining the calculated ratios. In the problem at hand, you're tasked with checking whether the given sequence is geometric:

• First compute the ratios between consecutive terms.
• Compare these ratios to see if they're identical.

If the ratios differ, as seen in the current example, the sequence is not geometric.
This step of analysis helps clarify the nature of the sequence and further supports understanding of key mathematical sequence properties.

College Algebra

In college algebra, understanding sequences is essential for grasping more complex mathematical concepts.
Sequences, including geometric ones, form the foundation for various algebraic problems and concepts in higher mathematics.

When solving algebra problems related to sequences, it's crucial to:

• Identify the type of sequence you are working with (geometric, arithmetic, etc.).
• Determine any key characteristics, such as the common ratio or common difference for arithmetic sequences.
• Apply these characteristics to solve problems or further analyze the sequence context.

In university-level algebraic contexts, understanding these sequences will aid in tackling problems involving exponential growth, functions, and series, bridging the gap between basic algebra and more advanced mathematical theory.