Problem 16 - Algebra and Trigonometry

Question

\(13-20\) . Determine whether the sequence is arithmetic. If is arithmetic, find the common difference. $$ 2,4,6,8, \dots $$

Step-by-Step Solution

Answer

Yes, it's arithmetic; common difference is 2.

Step 1: Identify the Sequence

The given sequence is 2, 4, 6, 8. To determine if it is arithmetic, we need to check the pattern between successive terms.

Step 2: Calculate Differences Between Terms

Subtract the first term from the second, the second from the third, and so on, to find the differences: \( 4 - 2 = 2 \) \( 6 - 4 = 2 \) \( 8 - 6 = 2 \).

Step 3: Verify Consistency of Differences

All differences are the same, which is 2. This means the sequence has a common difference and is arithmetic.

Step 4: Conclusion

The sequence is arithmetic because the differences between successive terms are consistent. The common difference is 2.

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is a key factor that helps determine whether a sequence follows the arithmetic pattern. The common difference is the constant amount that each term in the sequence is increased or decreased by to get to the next term. It is the hallmark of an arithmetic sequence. To find the common difference:

• Subtract any term in the sequence from the succeeding term. For example, in the sequence 2, 4, 6, 8, we subtract 2 from 4, 4 from 6, and so forth.
• If every calculated difference is the same, that value is called the common difference. In our example, each difference is 2, indicating this is indeed an arithmetic sequence.

This characteristic means that the differences are uniform across the sequence, thereby confirming its arithmetic nature. Understanding the common difference is essential to identify and predict future terms in the sequence.

Sequence Pattern

Recognizing a sequence pattern is crucial when determining the type of sequence. In an arithmetic sequence, a pattern emerges through the consistent application of the common difference. For the sequence 2, 4, 6, 8, observe the pattern:

• Each term increases by exactly 2 from the previous term. This repeatable addition creates a straightforward pattern.
• Such sequences are predictable because each term is constructed by adding the common difference to the previous term.

By identifying this repeating step, you understand that the arithmetic nature makes it easy to determine both past and future terms, providing a systematic way to describe and extend the sequence.

Successive Terms

Successive terms in an arithmetic sequence demonstrate the sequence's consistency and predictability. Each successive term is reached by adding the common difference to the preceding term. Consider these points:

• In the sequence 2, 4, 6, 8, the difference between each successive term and its previous term is constant, showing a distinct pattern.
• This consistency aids in forecasting future terms. For example, knowing the common difference of 2, the term succeeding 8 is 10.

This clarity in process and outcome is particularly useful for solving problems and predicting sequences, especially when you need to identify missing terms or extend the sequence further beyond what's provided.