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Chapter 1: Problem 12

Find the next term in list. \(0,5,10,15,20, \dots\)

Short Answer

Expert verified
The next term is 25.

Step by step solution

01

Identify the Pattern in the Sequence

The sequence given is \(0, 5, 10, 15, 20, \dots\). Observe the differences between consecutive terms. Calculate these differences: \(5 - 0 = 5\), \(10 - 5 = 5\), \(15 - 10 = 5\), \(20 - 15 = 5\). Each term increases by 5, indicating a pattern of addition.
02

Apply the Pattern to Find the Next Term

Since each term increases by 5, apply this pattern to the last known term, which is 20. Add 5 to 20: \(20 + 5 = 25\). This gives the next term in the sequence.

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Key Concepts

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

Consecutive Terms
In arithmetic sequences, consecutive terms are the numbers that appear one after the other in a sequence. It鈥檚 like a line-up where each number has a fixed position relative to its neighbors. In the sequence from the exercise, the numbers \(0, 5, 10, 15,\) and \(20\) are our consecutive terms. They appear consecutively, each following the other according to a specific rule.

Understanding how consecutive terms relate to each other is key to understanding the overall pattern of the sequence. If we take two consecutive terms and calculate the difference between them, it helps us see the regular step, referred to as the common difference, that guides the creation of the sequence.
Pattern Recognition
Recognizing patterns is an essential skill in mathematics, especially when dealing with sequences. Pattern recognition involves identifying the repeated process or rule that connects each term of a sequence. In the given sequence \(0, 5, 10, 15, 20\), you can see that each term increases by \(5\).

To spot the pattern, it helps to:
  • Look at the differences between consecutive terms, as shown in the step-by-step solution where the identified difference is a consistent \(5\).
  • Continue this process until you can confidently predict the next term by continuing the pattern you鈥檝e identified.
This pattern recognition provides the rule which the sequence follows, called the common difference in an arithmetic sequence.
Addition
Addition plays a central role in arithmetic sequences, particularly when the common difference is constant. Here, each term in the sequence is derived by adding the common difference to the previous term.

For example, in the sequence \(0, 5, 10, 15, 20\), you start with \(0\) and then add \(5\) to get the next term. This pattern continues with each step:
  • Add \(5\) to \(10\) to get \(15\).
  • Add \(5\) to \(15\) to get \(20\).
Continuing this method allows you to determine the next term after the last known term. In this case, adding \(5\) to \(20\) gives \(25\), the next term in the sequence.

Understanding addition in this context underlines how arithmetic sequences build systematically through simple repeatable operations.

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