Problem 61 - College Algebra with Modeling and Visualization

Question

Determine if \(f\) is an arithmetic sequence. $$f(n)=4-3 n^{3}$$

Step-by-Step Solution

Answer

No, \( f(n) = 4 - 3n^3 \) is not an arithmetic sequence.

Step 1: Understanding an Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This is known as the common difference.

Step 2: Evaluate Consecutive Terms

To determine if the sequence defined by \( f(n) = 4 - 3n^3 \) is arithmetic, we need to evaluate the first few terms:- \( f(1) = 4 - 3(1)^3 = 4 - 3 = 1 \)- \( f(2) = 4 - 3(2)^3 = 4 - 24 = -20 \)- \( f(3) = 4 - 3(3)^3 = 4 - 81 = -77 \)

Step 3: Calculate Differences Between Terms

Find the difference between successive terms:- Difference between \( f(2) \) and \( f(1) \) = \(-20 - 1 = -21\)- Difference between \( f(3) \) and \( f(2) \) = \(-77 - (-20) = -77 + 20 = -57\)

Step 4: Analyze the Differences

For \( f(n) \) to be an arithmetic sequence, the differences calculated in Step 3 must be the same. Since they are different (\(-21\) and \(-57\)), \( f(n) \) is not an arithmetic sequence.

Key Concepts

Common Difference

In the context of arithmetic sequences, the term "common difference" refers to the constant amount by which consecutive terms differ. This uniform difference is a defining characteristic of arithmetic sequences, allowing each term to be calculated by adding the common difference to the previous term. For example, in the sequence 2, 5, 8, 11, the common difference is 3. This means each term is 3 units greater than the term before it. Understanding the common difference is crucial because it determines the nature of the sequence. If the difference between the terms varies, as we observed in \( f(n) = 4 - 3n^3 \), then the sequence is not arithmetic. Recognizing consistency—or the lack of it—in differences helps identify the type of sequence you are dealing with. Without a common difference, as seen in the solution, the sequence defined by \( f(n) \) cannot be classified as arithmetic.

Consecutive Terms

When analyzing a sequence, evaluating consecutive terms is the first step to identify any patterns or rules, like the common difference in arithmetic sequences. Consecutive terms are simply terms that follow one another in the sequence without breaks or jumps. For instance, let's take a look at our sequence given by \( f(n) = 4 - 3n^3 \). By calculating the terms \( f(1) = 1 \), \( f(2) = -20 \), and \( f(3) = -77 \), we can specifically analyze how each term relates to its neighbors. In arithmetic sequences, the focus is especially on how each pair of consecutive terms is derived from the same common difference. However, as calculated in the example, the differences here are \( -21 \) and \( -57 \), which show inconsistency.Observing consecutive terms helps determine if the sequence structure aligns with any mathematical patterns, such as being arithmetic, geometric, or neither.

Sequence Evaluation

Sequence evaluation involves scrutinizing the setup of a sequence to determine its type and properties. This encompasses calculating terms, identifying the rules that generate the sequence, and verifying these against known criteria for sequence types, like arithmetic sequences.To evaluate the sequence given by \( f(n) = 4 - 3n^3 \), one must:

• Calculate several terms to see the progression.
• Find the differences between consecutive terms, as was done by calculating differences like \(-21\) and \(-57\).
• Check these differences for consistency, which indicates an arithmetic sequence.

The evaluation concluded that \( f(n) \) does not conform to an arithmetic pattern since the differences between consecutive terms are not uniform. Hence, sequence evaluation is key to understanding and categorizing different mathematical sequences based on their behavior and consistent properties.