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Chapter 1: Problem 1

Use inductive reasoning to determine the next element in each list. $$3,8,13,18,23$$

Short Answer

Expert verified
The next element is 28.

Step by step solution

01

- Identify the Pattern

First, observe the differences between consecutive terms. Calculate the differences: 8 - 3 = 5, 13 - 8 = 5, 18 - 13 = 5, 23 - 18 = 5.
02

- Determine the Common Difference

Notice that the difference between each consecutive term is 5. This suggests that the sequence is arithmetic with a common difference of 5.
03

- Apply the Pattern

To find the next term in the sequence, add the common difference to the last term. Calculate: 23 + 5 = 28.

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Key Concepts

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

arithmetic sequence
An arithmetic sequence is a list of numbers in which the difference between consecutive terms is always the same. This difference is known as the common difference. For example, consider the sequence given in the exercise: 3, 8, 13, 18, 23. If we look closely, we can see that each number increases by a consistent amount from the previous one.

This kind of sequence can be very useful in a variety of mathematical and real-world applications. Knowing how to identify and work with arithmetic sequences helps build a solid foundation for understanding more complex mathematical concepts.
common difference
The common difference is a critical element in arithmetic sequences. It is the value that, when added to each term, yields the next term in the sequence. In our example, to find the common difference, we subtract the first term from the second term (8 - 3 = 5) or any other successive terms:

  • 13 - 8 = 5
  • 18 - 13 = 5
  • 23 - 18 = 5
Thus, the common difference here is 5. This means that every term is 5 units greater than the previous term. By knowing this difference, one can easily find any term in the sequence by continually adding the common difference to the initial term(s).
pattern recognition
Pattern recognition is essential in sequences as it enables us to predict future terms based on observed systematic changes. To practice this skill, we carefully examine relationships or changes between terms.

Looking at the sequence 3, 8, 13, 18, 23, we identified that each term increases by 5. This recognition formed the basis for determining the common difference and thereby identifying the pattern. Having discerned this pattern, we can confidently state that the next term, as calculated by adding the common difference to the last term, is 28:
  • 23 + 5 = 28
Understanding patterns helps in problem-solving and making logical predictions in a range of mathematical contexts.

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Most popular questions from this chapter

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