Problem 19 - 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. $$16,9,2,-4, \dots$$

Step-by-Step Solution

Answer

No, the sequence is not arithmetic because the differences are not the same.

Step 1: Identify Differences

First, find the differences between consecutive terms in the sequence: \[ \text{Difference}_1 = 9 - 16 = -7 \] \[ \text{Difference}_2 = 2 - 9 = -7 \] \[ \text{Difference}_3 = -4 - 2 = -6 \]

Step 2: Check for Common Difference

Check if the differences calculated in Step 1 are all the same, which is a requirement for the sequence to be arithmetic. In the sequence, the differences are \(-7, -7,\) and \(-6\). The first two differences are the same, but the third difference is different.

Step 3: Conclusion on Arithmetic Sequence

Since the differences between the terms are not all the same, this sequence \(16, 9, 2, -4, \ldots\) is not an arithmetic sequence.

Key Concepts

Understanding Common Differences in Arithmetic Sequences

When dealing with arithmetic sequences, the idea of a **common difference** is vital. It determines whether a sequence is arithmetic or not. In an arithmetic sequence, each term after the first is obtained by adding a constant, known as the common difference, to the previous term. Consider a sequence: if you take the second term and subtract the first term, this gives you the common difference. You repeat this step for the third term minus the second, and the fourth term minus the third, and so on. For example, if these calculations all yield the same result, then that complement is the common difference, indicating that the sequence is indeed arithmetic. This is why consistency among these values is your key signal:

• The difference should be consistent across the sequence.
• A varying result indicates the sequence is not arithmetic.

Understanding common difference helps in predicting further terms in the sequence and knowing if the sequence falls in the arithmetic category.

Sequence Analysis: Is it Arithmetic?

Sequence analysis involves determining the nature of a sequence, primarily by examining the differences between terms, as done in arithmetic sequences. Here, sequence analysis is crucial in confirming if a sequence is arithmetic by checking the uniformity of the differences. When analyzing a sequence:

• Break down the sequence into individual terms.
• Calculate the differences between consecutive terms.
• Check the consistency among these differences.

This analysis tells us if a fixed number is being added as we move through the terms. If the differences remain consistent, you can deduce the sequence is arithmetic, meaning it has a regular interval that fits the definition of arithmetic sequences. If the pattern breaks, as it did in our example, where the differences were o -7, -7, and -6, then the sequence is not arithmetic. This small exercise in analyzing sequences helps bolster your understanding of regularity and predictability in arithmetic sequences.

Examining Differences Between Terms

The exploration of differences between terms is a basic yet powerful technique in identifying or ruling out arithmetic sequences. Here, let's dive into understanding this concept more deeply. To determine if a sequence is arithmetic, compute the differences between consecutive terms:

• Start by subtracting the first term from the second.
• Continue this process between all subsequent terms.

The goal here is to find a consistent trend. However, if any difference diverges from the established pattern, the sequence cannot be classified as arithmetic. In practice, clear understanding and keen observation of discrepancies in differences like -7, -7, and -6, alert us that the pattern falters after the third term, confirming that it’s not arithmetic. Examining differences helps you gain accuracy when deciphering sequences, and sharpens your skills in identifying underlying patterns. This approach is not just restricted to arithmetic sequences but is a useful analytic process in broader mathematical contexts.