What is the Next Number in the Series: 1, 5, 9, 13, 17, 21?

The sequence provided, 1, 5, 9, 13, 17, 21, is an arithmetic sequence. An arithmetic sequence is a series of numbers where each term increases (or decreases) by a constant difference known as the common difference. In this case, the common difference is 4. Understanding the pattern and the concept of an arithmetic sequence will help in identifying the next number in the sequence.

Identifying the Pattern

To determine the next number in the series, we first need to identify the common difference. You can verify this by checking the difference between consecutive terms in the sequence:

5 - 1 4 9 - 5 4 13 - 9 4 17 - 13 4 21 - 17 4

As can be seen, each term is 4 more than the previous term. This confirms that the common difference is indeed 4.

Applying the Common Difference

To find the next number in the sequence, we simply continue adding the common difference to the last term. The last term provided is 21. Therefore, the next number in the sequence is:

21 4 25

Arithmetic Sequence and Formulas

Arithmetic sequences can be expressed using a formula. If the first term of the sequence is denoted as (a) and the common difference as (d), the (n)-th term of the sequence can be calculated using the formula:

[a_n a (n-1)d]

For the sequence 1, 5, 9, 13, 17, 21, the first term (a 1) and the common difference (d 4). Plugging these values into the formula gives us:

[a_6 1 (6-1) times 4 1 20 21] and [a_7 1 (7-1) times 4 1 24 25]

Verifying the Pattern

Another way to verify the next number is by observing the pattern where each number is 4 more than the previous term. We can write out the sequence step by step:

1 4 5 5 4 9 9 4 13 13 4 17 17 4 21 21 4 25

Hence, the next number in the sequence is 25.

Conclusion

Understanding arithmetic sequences is crucial for solving problems involving patterns and progressions. In this case, the sequence 1, 5, 9, 13, 17, 21 follows a simple pattern where each term is 4 more than the previous term. By using this pattern or the arithmetic sequence formula, we can easily determine that the next number in the series is 25.

Additional Resources

If you want to learn more about arithmetic sequences and how to solve similar problems, you may find the following resources helpful:

Math Is Fun - Arithmetic Sequences Khan Academy - Arithmetic Sequence Practice