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Chapter 12: Problem 30

Is the sequence arithmetic? If so, identify the common difference. $$ 16,7,-2, \ldots $$

Short Answer

Expert verified
Yes, the sequence is arithmetic, and the common difference is -9.

Step by step solution

01

Identify the difference between consecutive terms

Subtract the first term from the second term, and the second term from the third term. In this case, 7 - 16 = -9 and -2 - 7 = -9.
02

Check if the difference is constant

Verify if the computed differences are the same. In this case, the difference is constant as -9 = -9.
03

Identify the common difference

If the difference is constant, then that difference is the common difference of the arithmetic sequence. In this case, the common difference is -9.

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Key Concepts

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

Common Difference
In an arithmetic sequence, the common difference is what sets it apart. It is the difference between consecutive terms, remaining constant all throughout the sequence. For instance, in the sequence given: \(16, 7, -2, \ldots\) the common difference is found by subtracting each term from the one that follows it. Begin with \(7 - 16 = -9\). Then confirm with the next set: \(-2 - 7 = -9\). Both resulted in \(-9\), indicating that this sequence is arithmetic, with \(-9\) being the common difference.

This common difference tells us how the sequence progresses from one term to the next, which is integral in predicting or computing any term in the sequence.
Sequence Analysis
When analyzing sequences, identifying the type of sequence is crucial. In the exercise, we analyzed an arithmetic sequence by confirming its common difference. Arithmetical sequences are defined by consistent changes between terms. Running through the sequence \(16, 7, -2, \ldots\), we checked consecutive terms and found equal differences, confirming its nature.

Analyzing sequences allows us to understand its pattern and predict other terms. For example:
  • Use the common difference to find subsequent terms.
  • Check if a term fits the sequence, maintaining the common difference.
  • Determine missing values, leveraging sequence patterns.
Through sequence analysis, it becomes easier to grasp a deeper understanding of mathematical patterns.
Algebra Concepts
Algebra plays a huge role in describing sequences; specifically, arithmetic sequences. Here, algebraic expressions help express terms in a sequence. For an arithmetic sequence with a common difference \(d\), any term \(a_n\) can be calculated using the formula:
\[ a_n = a_1 + (n-1) imes d \]
where \(a_1\) is the initial term, and \(n\) is the term number. This powerful algebra concept allows you to find any term without needing all preceding numbers.

Let's apply this with our sequence \(16, 7, -2, \ldots\):
  • The first term \(a_1\) is \(16\).
  • Common difference \(d\) is \(-9\).
  • To find the third term \(a_3\), apply the formula: \(16 + (3-1) imes (-9) = -2\).
This example perfectly demonstrates how algebra concepts simplify the methods of finding terms in arithmetic sequences, making calculations more straightforward.

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