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Chapter 1: Problem 24

Find the next three terms of each sequence. $$20,22,26,32, \ldots$$

Short Answer

Expert verified
The next three terms are 40, 50, and 62.

Step by step solution

01

Identify the pattern

Examine the differences between consecutive terms in the sequence: \(22 - 20 = 2\), \(26 - 22 = 4\), \(32 - 26 = 6\). The differences are increasing by 2 each time, suggesting the next difference should be \(8\).
02

Calculate the next term

Add the new difference, \(8\), to the last term, \(32\): \[32 + 8 = 40.\]Thus, \(40\) is the next term in the sequence.
03

Calculate the second next term

Continue with the next difference, which increases by 2 again. This would be \(8 + 2 = 10\). Add it to the last term, \(40\): \[40 + 10 = 50.\]Thus, \(50\) is the second next term.
04

Calculate the third next term

Proceed with the following difference, \(10 + 2 = 12\). Add this to the last term, \(50\): \[50 + 12 = 62.\]Thus, \(62\) is the third next term.

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Key Concepts

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

Sequence Patterns
A sequence is a set of numbers arranged in a specific order. Understanding the pattern allows us to predict future numbers in the series.
In an arithmetic sequence, each term after the first is the result of adding a fixed number called the "common difference" to the previous term. However, sequences can also have varying differences, like the one we've encountered here.
The sequence begins: 20, 22, 26, 32. At first glance, the numbers seem to be increasing gradually, but a deeper look reveals something more specific. The difference between each consecutive term is not fixed but changes consistently, increasing by 2 each time.
  • The difference between 22 and 20 is 2.
  • The difference between 26 and 22 is 4.
  • The difference between 32 and 26 is 6.
Consecutive Differences
Consecutive differences help uncover the underlying pattern in a sequence. By examining these differences, we can determine what number comes next.
The example sequence shows differences of 2, 4, and 6. These differences themselves form a pattern: they increase by 2. Recognizing this new pattern is crucial.
As we analyze these differences:
  • The first difference (22-20) is 2.
  • The second difference (26-22) is 4.
  • The third difference (32-26) is 6.
By spotting the changing consecutive differences, we extend the sequence. For the next term, we predict the difference will be 8 since 6 plus 2 equals 8.
Next Terms Calculation
Once you've identified the pattern and calculated the consecutive differences, you can easily find the next terms in the sequence.
Here's how to calculate the next terms:
  • Add the next difference, 8, to the last known term, 32, yielding 40.
  • To find the second next term, increase the previous difference by 2, making it 10, and add it to 40, resulting in 50.
  • Finally, for the third next term, increase the difference by 2 again, resulting in 12, and add it to 50, arriving at 62.
Using this method allows us to continue the sequence accurately: 32, 40, 50, and 62 as the next terms. You now know how to calculate future terms of this sequence by finding the new difference and adding it to the last term.

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