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Chapter 4: Problem 4

Tell how to find the next term in each pattern. \(20,26,32,38, \ldots\)

Short Answer

Expert verified
The next term is 44.

Step by step solution

01

Identify the Pattern

Examine the sequence of numbers: 20, 26, 32, 38. Observe that each term increases by the same amount compared to the previous term.
02

Determine the Common Difference

Calculate the difference between consecutive terms: 26 - 20 = 6, 32 - 26 = 6, 38 - 32 = 6. The common difference is 6.
03

Calculate the Next Term

To find the next term in the pattern, add the common difference to the last term: 38 + 6 = 44.
04

Write the Next Term

The next term in the sequence is 44.

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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 concept of a "common difference" is key. It is the constant amount by which each term in the sequence increases or decreases. Identifying this difference is a critical initial step in understanding the sequence.

To determine the common difference:
  • Look at two consecutive terms in the sequence.
  • Subtract the first term from the second term.
If you consistently get the same value from these subtractions, this value is your common difference. For example, in the sequence 20, 26, 32, 38, the difference is calculated as:
  • 26 - 20 = 6
  • 32 - 26 = 6
  • 38 - 32 = 6
Here, the common difference is 6. This tells us that each subsequent term is 6 units greater than the previous term.
Sequence Analysis
Analyzing an arithmetic sequence involves understanding how the sequence progresses. This means looking for patterns or consistencies in the order of the numbers. Finding such a pattern can help predict future terms.

In the given sequence - 20, 26, 32, 38 - we observe that:
  • Each term builds off the last by a specific increment, specifically the common difference of 6.
  • The sequence grows linearly, meaning it follows a straight-forward path of addition.
Analyzing the sequence in this manner helps to plan calculations for upcoming terms or even identify errors in an established pattern. Recognizing the linear growth due to a consistent common difference also sets arithmetic sequences apart from other mathematical sequences, like geometric sequences, which involve multiplication or division.
Term Calculation
Calculating the next term in an arithmetic sequence, once the common difference is known, is a simple process.

Using the sequence 20, 26, 32, 38, and knowing the common difference is 6, follow these steps:
  • Identify the last known term in the sequence, which is 38 in our example.
  • Add the common difference to this term: 38 + 6 = 44.
Thus, the next term in this sequence is 44. This straightforward mechanism shows how predictable arithmetic sequences are once the pattern is identified. This predictability allows for the efficient calculation of any further terms by repetitive addition of the common difference.

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