Question

tell whether the sequence is arithmetic. Explain your reasoning. \(\frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2}, \ldots\)

Short answer

Yes, the given sequence is arithmetic because the difference between each subsequent term is constant and equals \(\frac{1}{4}\).

Step-by-step solution

Identify the sequence

First of all, identify the given sequence, which is \(\frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2}, \ldots\)

Compute the differences

Now, compute the difference between subsequent terms. The difference between second term and first term is \(\frac{3}{4} - \frac{1}{2} = \frac{1}{4}\). Similarly, the difference between third term and second term is \(1 - \frac{3}{4} = \frac{1}{4}\), between fourth term and third term is \(\frac{5}{4} - 1 = \frac{1}{4}\), and between the fifth term and fourth term it's \(\frac{3}{2} - \frac{5}{4} = \frac{1}{4}\).

Check if differences are constant

Checking all acquired differences, it can be observed that the difference between each subsequent term is constant, equal to \(\frac{1}{4}\). Therefore, the given sequence \(\frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2}, \ldots\) is arithmetic.

Key concepts

Sequence Analysis

Understanding sequences involves recognizing patterns or rules that define the progression of numbers. In the given exercise, we are asked to analyze the sequence \( \frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2}, \ldots \). Sequence analysis helps us determine if the series of numbers follows specific properties.

An arithmetic sequence is identified by its property of having a common difference between consecutive terms. This means that if we subtract a term in the sequence from the next term, the result should always be the same value. Recognizing this key feature is crucial for determining if a sequence is arithmetic.

Therefore, analyzing the sequence involves examining these relationships between terms and ensuring consistency within its pattern. By doing this, we can decide not only if the sequence is arithmetic but also gain insights into its behavior.

Difference Calculation

Calculating the difference between sequence terms is an essential step to identify the nature of the sequence. To determine if a sequence is arithmetic, the difference between each term and the previous one must be constant. This fixed number, called the common difference, is central to the definition of arithmetic sequences.

In the given sequence, we subtract each term from the next:

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

Each calculated difference is \( \frac{1}{4} \), confirming that the sequence is indeed arithmetic.

This uniformity in differences is what defines the arithmetic sequence's structure and is a simple yet powerful tool to classify sequences.

Mathematical Reasoning

Mathematical reasoning forms the backbone of understanding and validating the properties of sequences. When working with arithmetic sequences, it is logical reasoning that confirms our findings.

In this exercise, we start with the basic concept of an arithmetic sequence and confirm it through computation. After performing difference calculations and establishing that each difference is \( \frac{1}{4} \), reasoning why they lead to an arithmetic sequence is important. Because sequences that share a constant difference between consecutive terms follow a linear pattern, they are classified as arithmetic.

This reasoned approach lays the groundwork for deeper mathematical understanding. By consistently applying these principles, students can develop strong problem-solving skills, which is vital in more complex mathematical contexts.