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Chapter 4: Problem 15

Find the next three terms of each sequence. \(5,9,13,17, \ldots\)

Short Answer

Expert verified
The next three terms are 21, 25, and 29.

Step by step solution

01

Identify the Pattern

First, let's observe how the sequence progresses. We see that each term increases by 4. Specifically, from 5 to 9, 9 to 13, and 13 to 17, the increase is consistently 4.
02

Calculate the Next Term

To find the next term after 17, add 4 to 17. So, \(17 + 4 = 21\). The next term is 21.
03

Calculate the Following Term

After finding 21, add 4 to this term to find the subsequent term. So, \(21 + 4 = 25\). The following term is 25.
04

Calculate the Final Term

Take the result of the previous step, which is 25, and add 4 to find the final term requested. So, \(25 + 4 = 29\). The final term is 29.

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Key Concepts

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

Pattern Recognition
Recognizing patterns is essential to understanding sequences. In the given sequence, each number increases by a consistent amount as you move from one term to the next. For example, from 5 to 9, and 9 to 13, you notice the difference is 4 every time. Detecting this regularity or pattern helps us identify the nature of the sequence.
In this case, because the difference between consecutive terms is a constant, this sequence is known as an *arithmetic sequence*. Recognizing such patterns allows us to predict future terms in the sequence based on the established rule.
Term Calculation
Once you've spotted the pattern, calculating additional terms becomes straightforward. Knowing that our sequence increases by 4 each time, finding the next terms is just a matter of addition.
For example:
  • To find the term after 17, simply add 4. Thus, \(17 + 4 = 21\).
  • Keep adding 4 to find successive terms: \(21 + 4 = 25\) and finally \(25 + 4 = 29\).
This method is systematic and can be applied to predict any number of future terms once the pattern is established.
Mathematical Progression
Understanding the idea of mathematical progression helps explain the nature of sequences. An arithmetic progression, like the one in our exercise, is a list of numbers where each term after the first is obtained by adding a fixed number, called the *common difference*, to the previous term.
In our specific example:
  • The first term (\( a = 5 \))
  • The common difference (\( d = 4 \))
can be used to determine any term in the sequence using the formula \(a_n = a + (n-1) \ imes d\), where \(a_n\) is the \(n^ ext{th}\) term. This formula ensures you can calculate terms far into the sequence without listing every preceding term.

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