Problem 15 - Precalculus Mathematics for Calculus

Question

Determine whether the sequence is arithmetic. If it is arithmetic, find the common difference. $$2,4,8,16, \dots$$

Step-by-Step Solution

Answer

The sequence is not arithmetic; it does not have a common difference.

Step 1: Identify Arithmetic Sequence Criteria

An arithmetic sequence has a constant difference between consecutive terms. This difference is called the 'common difference' and is denoted by \(d\).

Step 2: Calculate Differences between Terms

Calculate the difference between each consecutive pair of terms:- From 2 to 4: \(4 - 2 = 2\)- From 4 to 8: \(8 - 4 = 4\)- From 8 to 16: \(16 - 8 = 8\)

Step 3: Analyze Differences

Observe the calculated differences: 2, 4, and 8. The differences are not equal, indicating that the sequence does not have a common difference.

Step 4: Conclusion on Sequence Type

Since the differences between consecutive terms are not the same, the sequence is not arithmetic.

Key Concepts

Common Difference

To understand if a sequence is arithmetic, we need to identify the common difference. In simple terms, the common difference is the constant amount that is added to each term to get to the next term in an arithmetic sequence. For instance, if we have a sequence like 3, 6, 9, 12,..., the common difference is 3 because each number increases by 3 from the previous one.

When determining the common difference, it's crucial to ensure all consecutive terms have the same difference. If the differences fluctuate or are inconsistent, then the sequence cannot be classified as arithmetic. In our original exercise, checking the differences between terms showed us different values, meaning the sequence lacked a common difference.

Sequence Analysis

Sequence analysis is an important step in understanding sequences, whether arithmetic or not. It involves inspecting the sequence of numbers to discern a pattern or rule that governs it. This might be recognizing a recurrence relation, establishing a common difference, or identifying other patterns of regularity.

To analyze a sequence effectively:

• Identify the order and relation between the terms.
• Check for consistency in differences or ratios.
• Look for predictable patterns or benchmarks.

In the original exercise, sequence analysis involved calculating the differences between terms. By comparing these calculated differences, we can determine whether the sequence is arithmetic or not.

Consecutive Terms

Consecutive terms in a sequence follow one right after the other. In an arithmetic sequence, consecutive terms maintain a stable relationship through the common difference. Understanding how these terms relate is vital for sequence identification.

To examine consecutive terms:

• Look at how one term progresses to another.
• Calculate the difference between them.
• Ensure they're following a consistent pattern if an arithmetic sequence is claimed.

In our evaluation, from 2 to 4, then 4 to 8, and finally 8 to 16, there was no consistent pattern of change, showing these terms do not establish an arithmetic relationship.

Non-Arithmetic Sequence

A non-arithmetic sequence is a sequence in which there is no common difference between consecutive terms. This means the difference between terms is not consistent throughout the sequence. Non-arithmetic sequences can follow many different patterns - geometric, exponential, or even irregular sequences.

Recognizing a non-arithmetic sequence involves:

• Checking the differences or quotients between terms.
• Ensuring they do not match or remain constant.
• Looking for other types of patterns like multiplicative or non-linear trends.

In the original exercise, the differences of 2, 4, and 8 show variance, indicating the sequence is not arithmetic and might imply an alternative sequence type like geometric.