Problem 48 - Algebra 2

Question

Find the next four terms of each arithmetic sequence. \(1.3,3.8,6.3, \ldots\)

Step-by-Step Solution

Answer

The next four terms are 8.8, 11.3, 13.8, 16.3.

Step 1: Identify the Common Difference

An arithmetic sequence is a sequence of numbers with a constant difference between consecutive terms. To find the common difference, subtract the first term from the second term. Here, the first term is 1.3 and the second term is 3.8. The common difference is calculated as follows:\[ d = 3.8 - 1.3 = 2.5 \]Thus, the common difference is 2.5.

Step 2: Calculate the Fourth Term

With the common difference identified, we can use it to calculate the next term of the sequence. The fourth term is found by adding the common difference to the third term. The third term is 6.3:\[ 6.3 + 2.5 = 8.8 \]Thus, the fourth term is 8.8.

Step 3: Calculate the Fifth Term

To find the fifth term, add the common difference to the fourth term:\[ 8.8 + 2.5 = 11.3 \]Thus, the fifth term is 11.3.

Step 4: Calculate the Sixth Term

Next, add the common difference to the fifth term to find the sixth term:\[ 11.3 + 2.5 = 13.8 \]Thus, the sixth term is 13.8.

Step 5: Calculate the Seventh Term

Finally, add the common difference to the sixth term to find the seventh term:\[ 13.8 + 2.5 = 16.3 \]Thus, the seventh term is 16.3.

Key Concepts

Understanding the "Common Difference"

In an arithmetic sequence, the magic number that helps us connect each term to the next one is called the "common difference." It is the amount you need to add to one term to get to the next consecutive term.
This value is consistent throughout the entire sequence, meaning it doesn't change as we move through the terms.

To find the common difference in a sequence:

• Select two consecutive terms — for instance, the first and second terms.
• Subtract the earlier term from the later term.
  In our example sequence, we subtract the first term 1.3 from the second term 3.8, yielding a common difference of 2.5: \( d = 3.8 - 1.3 = 2.5 \)

Identifying the common difference is essential because it forms the backbone of the sequence's progression. With it, you can foresee all upcoming terms by repeated addition.

What is a "Sequence of Numbers"?

A sequence of numbers is essentially an ordered list of numbers arranged according to a specific rule or pattern. In mathematics, we often encounter different types of sequences, but an arithmetic sequence follows a particularly straightforward rule: it builds by adding a fixed number — known as the common difference — to each preceding term.

This sequence is crucial in various aspects of mathematics and can often appear in real-world scenarios, such as calculating interest or analyzing patterns.
Each term in a sequence is generated based on its position and the relation to the previous terms.

• The sequence given, starting with 1.3, 3.8, 6.3, follows the rule of adding 2.5 to find subsequent numbers.
• This systematic approach makes predicting the next terms straightforward: 8.8, 11.3, 13.8, and 16.3.

Understanding sequences helps break down complex problems into manageable steps by simplifying them through predictable patterns.

Exploring "Consecutive Terms"

In a sequence, consecutive terms refer to numbers that appear one right after another without any interruptions in between. These terms are arranged based on a rule or pattern that the sequence follows.

In an arithmetic sequence, consecutive terms are generated by applying the common difference repeatedly.

• This ensures that the same numerical step is drawn from one term to the next.
• For instance, examining our sequence (1.3, 3.8, 6.3), each term comes right after another and is attained by adding the common difference of 2.5.

By understanding the nature of consecutive terms, you can efficiently establish a rhythm to unravel the entire sequence. Recognizing these terms' linkage clarifies how the entire sequence continues to evolve.