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Chapter 11: Problem 62

Determine whether each given sequence could be an arithmetic sequence. $$-7,-4,-2,0, \dots$$

Short Answer

Expert verified
No, it is not an arithmetic sequence.

Step by step solution

01

Identify the Terms in the Sequence

The given terms in the sequence are $$a_1 = -7, a_2 = -4, a_3 = -2, a_4 = 0$$.
02

Calculate the Differences Between Consecutive Terms

Find the difference between each consecutive term: $$a_2 - a_1 = -4 - (-7) = -4 + 7 = 3$$ $$a_3 - a_2 = -2 - (-4) = -2 + 4 = 2$$ $$a_4 - a_3 = 0 - (-2) = 0 + 2 = 2$$
03

Determine if the Common Difference is Consistent

Observe the differences calculated:The difference between the first and second terms is 3.The difference between the second and third terms is 2.The difference between the third and fourth terms is 2.Since the differences are not the same, the sequence does not have a common difference.
04

Conclusion

The sequence $$-7, -4, -2, 0, \text{...}$$ does not have a consistent common difference, so it is not an arithmetic sequence.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common Difference
In arithmetic sequences, the most important idea is the common difference. This is the amount you add to one term to get the next term. For example, if you start with a number and keep adding 3 each time, then 3 is your common difference. You can find the common difference in a sequence by subtracting a term from the one that follows it. For instance, in the sequence \(-7, -4, -2, 0\), the common difference between the first and second term would be calculated as \(-4 - (-7) = 3\). However, not all terms have this difference, meaning this sequence is not arithmetic.
Sequence Terms
Sequence terms are the individual numbers in a sequence. In the given exercise, the sequence terms are \(-7, -4, -2, 0\). When working with sequences, you often need to identify the terms first. Each term in a sequence has a specific position. For example, \(-7\) is the first term \(a_1\) and \(-4\) is the second term \(a_2\). It鈥檚 crucial to understand the place of each term to analyze the sequence correctly.
Consistent Differences
Consistent differences in a sequence are a sign of an arithmetic sequence. This means the difference between consecutive terms should be the same throughout the entire sequence. In the example sequence \(-7, -4, -2, 0\), the differences were calculated as \3, 2,\ and \2\. Because these differences are not consistent, the sequence cannot be classified as arithmetic. For a sequence to be arithmetic, all these differences must match.
Precalculus
Precalculus includes the study of sequences and series, preparing you for higher-level mathematics. Understanding how sequences like arithmetic sequences work is a foundational skill. Mastery of the concepts such as common difference and consistent differences will help you solve more complex problems in calculus and beyond. Grasping these basic concepts ensures that you have the tools needed to progress smoothly in mathematics.

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