Question

Find the next three terms of each sequence. $$1,3,5,7, \ldots$$

Short answer

The next three terms are 9, 11, and 13.

Step-by-step solution

Identify the Pattern

Observe the sequence: 1, 3, 5, 7. Notice that each term is increasing by 2 from the previous term. Hence, this sequence is an arithmetic sequence with a common difference of 2.

Determine the Next Term

To find the next term after 7, add the common difference, which is 2, to the last known term: \[ 7 + 2 = 9 \] Therefore, the first of the next three terms is 9.

Calculate the Following Term

Continue by adding the common difference, 2, to the latest term found, which is 9: \[ 9 + 2 = 11 \] So, the second of the next three terms is 11.

Find the Last Term in the Series

Add the common difference, 2, to the term we just found, which is 11: \[ 11 + 2 = 13 \] Thus, the third of the next three terms is 13.

Key concepts

Common Difference

When dealing with arithmetic sequences, one of the most significant elements to understand is the **common difference**. This term refers to the fixed amount between each consecutive number in the sequence. In simpler terms, it's what you add to one term to get to the next one. For example, looking at the sequence 1, 3, 5, 7, we notice that by adding 2 to each term, we reach the next number. Hence, the common difference here is 2.

• Identifying the common difference helps you easily find future terms.
• Think of it as a consistent step size as you walk through the sequence.

Getting comfortable with finding the common difference allows you to master the basics of arithmetic sequences and solve related problems with ease.

Next Terms

Once you've identified the common difference in a sequence, finding the **next terms** becomes straightforward. The task involves simply adding the common difference to the latest known term. Let's apply this concept to our sequence:

• The last given term is 7. By adding the common difference of 2, the first next term is 9: 7 + 2 = 9.
• Continuing with this process, add 2 to 9 to get the next term: 9 + 2 = 11.
• Finally, add 2 to 11 to find the final next term: 11 + 2 = 13.

Knowing how to calculate the next terms will empower you to extend sequences effortlessly. This process relies solely on arithmetic addition, which makes it a simple but powerful technique.

Number Sequences

**Number sequences** are collections of numbers that follow a particular pattern or rule. In the case of arithmetic sequences, the sequence results from consistently adding a common difference to the terms. They are fundamentally about patterns and predictability.

• An arithmetic sequence is a classic example, defined by its constant common difference.
• Recognizing these patterns helps in predicting unknown terms without needing all previous numbers.

Understanding number sequences is crucial because they form the backbone of various mathematical concepts. By mastering these foundations, you can solve complex problems and notice patterns in numbers, unlocking a deeper appreciation of mathematics.