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Chapter 1: 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, 29.

Step by step solution

01

Identify the Pattern

Notice the sequence: 5, 9, 13, 17. We need to determine the pattern in this sequence. Each term is obtained by adding 4 to the previous term. Thus, the difference between consecutive terms is 4.
02

Apply the Pattern to Find the Next Term

To find the next term, add 4 to the last known term in the sequence, which is 17: 17 + 4 = 21. Therefore, the next term is 21.
03

Apply the Pattern Again for the Second Next Term

Add 4 to the next term we found in Step 2, which is 21: 21 + 4 = 25. Therefore, the term after 21 is 25.
04

Find the Third Next Term

Add 4 to the last term obtained, which is 25: 25 + 4 = 29. Therefore, the third next 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
In mathematics, pattern recognition is a crucial skill that helps us identify regularities or trends within a set of data points or terms. When dealing with sequences, recognizing the pattern is the first step to predicting future terms.
For the sequence 5, 9, 13, 17, we start by observing the differences between consecutive numbers.
By calculating
  • 9 - 5 = 4
  • 13 - 9 = 4
  • 17 - 13 = 4
we notice a consistent pattern: each term increases by 4.
Recognizing this repetitive addition helps us identify the sequence as arithmetic, allowing us to extend it by continually applying the same increment.
Sequence
A sequence is an ordered list of numbers connected by a specific rule or pattern. In sequences like the one 5, 9, 13, 17, the numbers are arranged in a definite order where each number is called a "term."
There are many types of sequences, such as arithmetic, geometric, and Fibonacci, each following a unique rule.
The type of sequence determines how each term relates to its predecessors.
In an arithmetic sequence, each term is found by adding a constant value to the previous term.
The "rule" of the sequence allows us to predict additional terms and understand the pattern of the numbers.
Arithmetic Progression
Arithmetic progression is a type of sequence where the difference between consecutive terms is always constant. This difference is known as the "common difference."
For the sequence 5, 9, 13, 17, the common difference is 4, as we add 4 to each term to get to the next.
  • The formula for the nth term of an arithmetic sequence is: \[ a_n = a_1 + (n-1) imes d \] where
    • \( a_n \) is the nth term,
    • \( a_1 \) is the first term,
    • \( n \) is the term number,
    • \( d \) is the common difference.
Understanding arithmetic progression helps us not only predict the next terms but also calculate any term in the sequence without having to list all prior terms.

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