Problem 23 - Precalculus: Mathematics for Calculus

Question

The first four terms of a sequence are given. Can these terms be the terms of an arithmetic sequence? If so, find the common difference. $$3, \frac{3}{2}, 0,-\frac{3}{2}, \dots$$

Step-by-Step Solution

Answer

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

Step 1: Define an Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant is called the common difference, denoted by \(d\).

Step 2: Check Consecutive Differences

To determine if the given sequence \(3, \frac{3}{2}, 0, -\frac{3}{2}\) is arithmetic, compute the difference between consecutive terms: \(a_2 - a_1\), \(a_3 - a_2\), and \(a_4 - a_3\). If all these differences are the same, then the sequence is arithmetic.

Step 3: Calculate Differences

Calculate each difference: \((\frac{3}{2} - 3) = -\frac{3}{2}\), \((0 - \frac{3}{2}) = -\frac{3}{2}\), \(( -\frac{3}{2} - 0 ) = -\frac{3}{2}\).

Step 4: Verify Commonality of Differences

Check if all calculated differences are equal. Here, all differences are \(-\frac{3}{2}\). Therefore, the sequence is an arithmetic sequence.

Step 5: State the Common Difference

Since each difference is \(-\frac{3}{2}\), the common difference \(d\) of the sequence is \(-\frac{3}{2}\).

Key Concepts

Common Difference

In an arithmetic sequence, the term **common difference** plays a crucial role. It is the value that separates each term in the sequence from the one that follows it. To determine this value, simply subtract the first term from the second term.
The idea is to have a consistent gap between the terms. For instance, in the sequence presented as part of the exercise, these terms are: 3, \( \frac{3}{2} \), 0, and \( -\frac{3}{2} \). The common difference, denoted by \( d \), can be verified by calculating \( a_2 - a_1 \), \( a_3 - a_2 \), and \( a_4 - a_3 \).

• For the first two terms, the difference is \( \frac{3}{2} - 3 = -\frac{3}{2} \).
• Between the second and third terms, it's \( 0 - \frac{3}{2} = -\frac{3}{2} \).
• And, from the third to the fourth term, the difference remains \( -\frac{3}{2} \).

Notice that all of these calculations yield the same number \( -\frac{3}{2} \). This uniformity confirms that the common difference of the sequence is \( -\frac{3}{2} \). Understanding the common difference is fundamental as it defines the regularity of the sequence.

Consecutive Terms

When examining an arithmetic sequence, focusing on **consecutive terms** is essential. Consecutive terms are those that come one after another in the sequence. These terms help us to understand the structure of the sequence by spotting the repetitive pattern defined by the common difference.
To analyze consecutive terms effectively, take two neighboring terms in a sequence and determine their difference. This trick is especially useful when trying to verify if a sequence is arithmetic.
In the example sequence 3, \( \frac{3}{2} \), 0, and \( -\frac{3}{2} \), each pair of consecutive terms is separated by the same amount, which is \( -\frac{3}{2} \). This consistent difference across consecutive terms validates the arithmetic nature of the sequence, which comprises a simple pattern: *each next term is \( -\frac{3}{2} \) less than the one before*.By mastering the recognition of patterns in consecutive terms, one can easily delve into the characteristic of any arithmetic sequence swiftly.

Sequence Analysis

**Sequence analysis** involves scrutinizing a set of numbers to determine any underlying patterns or regularities. For arithmetic sequences, this analysis centers on establishing whether a constant common difference exists between every pair of consecutive terms.
Sequence analysis follows these broad steps:

• Identify the sequence and understand its terms.
• Calculate the difference between each pair of consecutive terms.
• Verify that these differences are uniform.
• Conclusively determine whether the sequence is arithmetic.

In our case, the sequence provided: 3, \( \frac{3}{2} \), 0, \( -\frac{3}{2} \), ... successfully satisfied all these conditions. Each step was methodically executed to affirm that an arithmetic relationship holds across terms consistently.
Engaging in sequence analysis allows students not just to confirm the nature of a sequence but also gain deeper insights into its progression and potential uses, such as predicting future terms or solving complex mathematical problems with ease.