Problem 7 - College Algebra

Question

For the following exercises, find the common difference for the arithmetic sequence provided. $$ \left\\{0, \frac{1}{2}, 1, \frac{3}{2}, 2, \ldots\right\\} $$

Step-by-Step Solution

Answer

The common difference is \( \frac{1}{2} \).

Step 1: Identify Consecutive Terms

To find the common difference in an arithmetic sequence, first identify two consecutive terms in the sequence. In this sequence, the first two terms are 0 and \( \frac{1}{2} \).

Step 2: Subtract the First Term from the Second Term

Calculate the difference between the second term and the first term: \( \frac{1}{2} - 0 = \frac{1}{2} \).

Step 3: Verify with Subsequent Pairs

To confirm the common difference, check the difference for another pair of consecutive terms:\(1 - \frac{1}{2} = \frac{1}{2} \)This confirms that the common difference is consistent.

Key Concepts

Understanding the Common Difference

The common difference is a crucial element in an arithmetic sequence. It represents the constant amount we add (or subtract) between each pair of consecutive terms. For an arithmetic sequence, all pairs of consecutive terms have this exact difference. It essentially dictates the rate at which the sequence progresses, making it a key feature to recognize.

To find the common difference, follow these simple steps:

• Pick any two consecutive terms in the sequence.
• Subtract the first term of the pair from the second term.
• If needed, repeat this with another pair to ensure consistency throughout the sequence.

For instance, in the sequence \( \{0, \frac{1}{2}, 1, \frac{3}{2}, 2, \ldots\} \), the common difference is \( \frac{1}{2} \), obtained by calculating \( \frac{1}{2} - 0 \). Thus, each term is derived by adding \( \frac{1}{2} \) to the previous term.

Identifying Consecutive Terms

Consecutive terms in a sequence are terms that appear right after one another. In simple terms, they are next-door neighbors in the sequence. When working with arithmetic sequences, identifying consecutive terms will help to easily determine the common difference.

For example, given a sequence \( \{0, \frac{1}{2}, 1, \frac{3}{2}, 2, \ldots\} \), the terms \( 0 \) and \( \frac{1}{2} \) are consecutive, as are \( \frac{1}{2} \) and \( 1 \), and so on. You can select any adjacent pair to investigate the sequence's properties, as the difference will remain constant due to the arithmetic nature of the sequence.

By analyzing consecutive terms step by step, one can ensure that the calculated common difference is consistent, confirming that the sequence is indeed arithmetic. This method not only helps to find the common difference but also guides you in verifying the sequence's correctness.

The Process of Sequence Analysis

Sequence analysis involves understanding the structure and pattern of a sequence. In arithmetic sequences like the one we are examining, you'll focus on the common difference as a key characteristic.

When conducting sequence analysis:

• Begin by reviewing the sequence and noting down consecutive terms.
• Calculate the common difference as discussed previously.
• Verify the difference by comparing multiple consecutive term pairs to ensure consistency throughout the sequence.

Through sequence analysis, you not only determine the common difference but can also predict future terms in the sequence by continuously applying the common difference. This analysis transforms the sequence into a powerful tool for solving various mathematical problems and understanding the sequence's progression, making it an essential technique for students to master.