Problem 9 - Algebra and Trigonometry

Question

9–16 Determine whether the sequence is arithmetic. If it is arithmetic, find the common difference. $$5,8,11,14, \dots$$

Step-by-Step Solution

Answer

Yes, it is arithmetic with a common difference of 3.

Step 1: Identify the Sequence

The sequence given is \(5, 8, 11, 14, \dots\). We need to determine if this is an arithmetic sequence.

Step 2: Find the Differences

To check if the sequence is arithmetic, calculate the difference between consecutive terms: \(8 - 5 = 3\), \(11 - 8 = 3\), \(14 - 11 = 3\).

Step 3: Determine if the Differences are Constant

The differences between consecutive terms are all \(3\). Since the differences are constant, the sequence is arithmetic.

Step 4: Identify the Common Difference

In an arithmetic sequence, the common difference is the constant difference between terms. Here, the common difference is \(3\).

Key Concepts

Common Difference

In arithmetic sequences, the common difference is key to understanding how the sequence progresses. It is defined as the constant value that you add to each term to get to the next term.
Let's break this down: If you have a sequence such as 5, 8, 11, 14, and so on, you find the common difference by subtracting one term from the next. For instance:

• Subtract 5 from 8: 8 - 5 = 3
• Subtract 8 from 11: 11 - 8 = 3
• Subtract 11 from 14: 14 - 11 = 3

As you can see, each difference is the same. This constant difference of 3 tells us not only that this is an arithmetic sequence but also how it grows. The common difference helps to identify where the sequence will go next, making it a powerful tool for predicting future terms.

Consecutive Terms

Understanding consecutive terms in a sequence is crucial for spotting patterns like those in arithmetic sequences. Consecutive terms are simply terms that follow one another in order.
In the sequence 5, 8, 11, 14, each term is the next in line. They are 'consecutive' because there are no terms skipped in between them. When identifying properties of sequences, you examine consecutive terms to find the common difference.
For example, by looking at 5 and 8, or 8 and 11, you're examining consecutive terms to determine how each term relates to the next. This straightforward observation seals the identification of the sequence type, particularly when they share a consistent difference.

Sequence Identification

To identify an arithmetic sequence, you examine the differences between consecutive terms to see if they are consistent. This means checking if the same number is being added or subtracted to get from one term to the next.
Let's review the sequence 5, 8, 11, 14. We see that each transition from one number to the next involves adding 3. Identifying sequences involves steps:

• Observe the pattern of numbers.
• Calculate differences between consecutive terms.
• Check for consistency in those differences.

If the differences are consistent, as they are in this example, the sequence is arithmetic. Successfully identifying this can help predict future terms, just by continuing to add the common difference.