Problem 9 - Precalculus with Limits

Question

In Exercises 5 - 14, determine whether the sequence is arithmetic. If so, find the common difference. \( \dfrac{9}{4}, 2, \dfrac{7}{4}, \dfrac{3}{2}, \dfrac{5}{4}, \cdots \)

Step-by-Step Solution

Answer

Yes, the sequence is arithmetic with a common difference of \(-\frac{1}{4}\).

Step 1: Understand the arithmetic sequence

An arithmetic sequence has a common difference between consecutive terms. The general form of an arithmetic sequence is \(a, a + d, a + 2d, a + 3d, \cdots\), where \(a\) is the first term and \(d\) is the common difference.

Step 2: Compute the differences

To find whether the given sequence is arithmetic, calculate the differences between consecutive terms: \(2 - \frac{9}{4} = -\frac{1}{4}\), \(\frac{7}{4} - 2 = -\frac{1}{4}\), \(\frac{3}{2} - \frac{7}{4} = -\frac{1}{4}\), \(\frac{5}{4} - \frac{3}{2} = -\frac{1}{4}\). The computed differences are the same, which means it is an arithmetic sequence.

Step 3: Identify the common difference

Since the differences are the same, the common difference of the sequence is \(-\frac{1}{4}\).

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is a vital concept that defines the sequence's uniformity. It refers to the consistent amount added or subtracted to get from one term to the next. This value is denoted by the symbol \(d\), and its presence means the sequence maintains a specific pattern or progression.

To visualize this, consider an arithmetic sequence like \(a, a + d, a + 2d, a + 3d, \ldots \). Here, \(a\) is the first term, and each subsequent term is determined by adding the common difference \(d\). Let's apply this to understand better:

• Given sequence: \( \frac{9}{4}, 2, \frac{7}{4}, \frac{3}{2}, \frac{5}{4}, \ldots \)
• By calculating: \(2 - \frac{9}{4} = -\frac{1}{4}\); apply similarly to other terms.

The consistent \(-\frac{1}{4}\) difference among all consecutive terms confirms an arithmetic sequence, with \(d = -\frac{1}{4}\).

Understanding the common difference helps recognize patterns and predict future sequence terms.

Consecutive Terms

Consecutive terms in an arithmetic sequence are terms that appear one after another, following the sequence's established order. Each term is generated by applying the common difference to the preceding term. These terms are crucial to understand the sequence's flow and ensure it adheres to the arithmetic pattern.

Consider the concept applied to our example:

• First term: \( \frac{9}{4} \)
• Second term: \(2\)
• Third term: \( \frac{7}{4} \)
• and so forth.

Upon examination, each term subtracts a common difference of \(-\frac{1}{4}\) from the previous term. This logical progression confirms the arithmetic nature.

Using consecutive terms, you can assess whether you are following the arithmetic sequence rules. Note how each term derives from its predecessor, validating the sequence's consistency with its initial setup.

Sequence Analysis

Sequence analysis is the process of examining a sequence to determine its type and the relationships between its terms. Analyzing sequences can reveal overarching patterns or specific traits, such as being arithmetic, geometric, or neither.

When analyzing the given sequence \( \frac{9}{4}, 2, \frac{7}{4}, \frac{3}{2}, \frac{5}{4}, \ldots \), start by exploring the differences between each pair of consecutive terms. You calculate:

• \( 2 - \frac{9}{4} = -\frac{1}{4} \)
• \( \frac{7}{4} - 2 = -\frac{1}{4} \)
• \( \frac{3}{2} - \frac{7}{4} = -\frac{1}{4} \)
• \( \frac{5}{4} - \frac{3}{2} = -\frac{1}{4} \)

Each calculation results in the same difference, \(-\frac{1}{4}\). This uniformity reveals the series as arithmetic. Sequence analysis showcases how critical it is to calculate differences and spot consistent patterns throughout.

Such assessments significantly enhance your understanding of sequences and their defining characteristics. Employ sequence analysis to gain insights into the structure and predictability of numerical progressions.