Problem 13 - Algebra and Trigonometry

Question

9–16 Determine whether the sequence is arithmetic. If it is arithmetic, find the common difference. $$3, \frac{3}{2}, 0,-\frac{3}{2}, \dots$$

Step-by-Step Solution

Answer

The sequence is arithmetic with a common difference of \(-\frac{3}{2}\).

Step 1: Identify Sequence Terms

First, let's write out the first few terms of the sequence: \(3, \frac{3}{2}, 0, -\frac{3}{2}, \ldots\).

Step 2: Calculate Differences Between Consecutive Terms

Subtract each term from the next term to determine the differences: - Difference between the first and second term: \( \frac{3}{2} - 3 = \frac{3}{2} - \frac{6}{2} = -\frac{3}{2}\) - Difference between the second and third term: \(0 - \frac{3}{2} = -\frac{3}{2}\) - Difference between the third and fourth term: \(-\frac{3}{2} - 0 = -\frac{3}{2}\).

Step 3: Determine if Sequence is Arithmetic

A sequence is arithmetic if the difference between consecutive terms (called the common difference) is constant. From Step 2, we found that the differences are \(-\frac{3}{2}\), \(-\frac{3}{2}\), and \(-\frac{3}{2}\). This consistent difference indicates that the sequence is arithmetic.

Step 4: Identify The Common Difference

Since the differences are consistent, the common difference for this arithmetic sequence is \(-\frac{3}{2}\).

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is the consistent amount that each term changes from one to the next. You can think of it as the 'step' you take to move from one number to its successor in the sequence. This difference is crucial for determining if a sequence is arithmetic or not. By simply subtracting any term from the one that follows it, you can investigate if the sequence maintains this consistent pattern. For example, let's consider the sequence: \(3, \frac{3}{2}, 0, -\frac{3}{2}, \ldots\) When we look at this sequence, we observe the following differences:

• Between the first and second terms: \(\frac{3}{2} - 3 = -\frac{3}{2}\)
• Between the second and third terms: \(0 - \frac{3}{2} = -\frac{3}{2}\)
• Between the third and fourth terms: \(-\frac{3}{2} - 0 = -\frac{3}{2}\)

Every difference amounts to \(-\frac{3}{2}\), hence this sequence has a constant common difference.

Consecutive Terms

Consecutive terms are simply the terms that appear one after the other in a sequence. Understanding consecutive terms helps to highlight the regularity or pattern within a sequence. Let's take a closer look. In a sequence like \(3, \frac{3}{2}, 0, -\frac{3}{2}, \ldots\), each term follows and relates to the term before it in a unique way based on the common difference. We can identify these terms as:

• First term: \(3\)
• Second term: \(\frac{3}{2}\)
• Third term: \(0\)
• Fourth term: \(-\frac{3}{2}\)

By comparing the consecutive terms, we're able to spot the common difference.
This negative step of \(-\frac{3}{2}\) between each pair of terms confirms that the sequence is not only arithmetic, but also decreasing consistently.

Sequence Determination

Sequence determination involves deciding whether a sequence is arithmetic or not based on the differences between its consecutive terms. The hallmark of an arithmetic sequence is the constant difference between consecutive terms.To determine if a sequence like \(3, \frac{3}{2}, 0, -\frac{3}{2}, \ldots\) is arithmetic, we calculate the differences between consecutive terms and check for consistency. From our calculations, the differences

• \(\frac{3}{2} - 3 = -\frac{3}{2}\)
• \(0 - \frac{3}{2} = -\frac{3}{2}\)
• \(-\frac{3}{2} - 0 = -\frac{3}{2}\)

remain constant. This consistent common difference of \(-\frac{3}{2}\) indicates that the sequence is indeed arithmetic.
Thus, determining the sequence type helps in understanding the pattern and predicting future terms efficiently.