Question

tell whether the sequence is arithmetic. Explain your reasoning. \(\frac{1}{6}, \frac{1}{2}, \frac{5}{6}, \frac{7}{6}, \frac{3}{2}, \ldots\)

Short answer

No, the sequence is not arithmetic because the differences between terms are not constant.

Step-by-step solution

Find the difference between the first two terms

Subtract the first term from the second term. So, \( \frac{1}{2} - \frac{1}{6} = \frac{4}{6} \) or \( \frac{2}{3} \)

Find the difference between the second and the third terms

Repeat the process with the second and third term in the sequence. So, \( \frac{5}{6} - \frac{1}{2} = \frac{4}{6} \) or \( \frac{2}{3} \)

Compare the differences

By checking the results, the differences are the same: \( \frac{2}{3} \) in both cases.

Verify with more terms

To be sure, repeat the process with more terms if needed. In this case, calculating \( \frac{7}{6} - \frac{5}{6} \) and \( \frac{3}{2} - \frac{7}{6} \), both yield \( \frac{2}{6} \) or \( \frac{1}{3} \). These are not equal to the common difference \( \frac{2}{3} \) found before.

Conclusion

Because the differences are not constant throughout the sequence, it can be concluded that this is not an arithmetic sequence.

Key concepts

Sequence Differences

When exploring arithmetic sequences, understanding sequence differences is essential. A sequence is a set of numbers arranged in a specific order. To find out if a sequence is arithmetic, you'll need to calculate the difference between consecutive terms. This difference, when consistent, signals an arithmetic sequence.

• Begin by subtracting the first term from the second term.
• Then, subtract the second term from the third term, and so on.

This allows you to identify if each of these differences are uniform across the terms. In arithmetic sequences, these differences should be identical. Always check multiple pairs to ensure accuracy.

Common Difference

The term 'common difference' refers to the consistent disparity between consecutive numbers in an arithmetic sequence. It's the key feature that characterizes these sequences. When the common difference is the same throughout, the sequence is indeed arithmetic.

• Example: If the first term is \( rac{1}{6} \) and the second term is \( rac{1}{2} \), the common difference is \( rac{2}{3} \).
• If this is maintained, like from \( rac{1}{2} \) to \( rac{5}{6} \), again \( rac{2}{3} \), you are on track.

But, if later differences deviate, as seen in \( rac{7}{6} - rac{5}{6} \) which results in \( rac{1}{3} \), the sequence breaks its arithmetic pattern.
Knowing this common difference helps with predictions of future sequence numbers as well.

Reasoning in Sequences

Reasoning in sequences involves a careful examination of patterns and consistencies. To determine if a sequence is arithmetic, or if it could potentially follow another mathematical form, you must rely on logical steps and evidence.
This process begins with identifying each pair's difference. If these differences match consistently, you have an arithmetic sequence. However, when they vary - as in our original sequence - it indicates a break from the arithmetic pattern. With discrepancies in differences, you recognize that another form of sequence or inconsistency is in play.
In mathematical reasoning, it's essential to consistently examine and verify until the pattern becomes apparent. This often involves revisiting initial terms and ensuring accurate calculations, providing a clear insight into what type of sequence you are examining.