Problem 26 - Precalculus: Mathematics for Calculus

Question

The first four terms of a sequence are given. Can these terms be the terms of an arithmetic sequence? If so, find the common difference. $$\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \dots$$

Step-by-Step Solution

Answer

No, these terms cannot form an arithmetic sequence as the differences are not constant.

Step 1: Identify the Terms

The given terms of the sequence are \(\frac{1}{2}\), \(\frac{1}{3}\), \(\frac{1}{4}\), and \(\frac{1}{5}\). An arithmetic sequence requires that the difference between each consecutive term be constant.

Step 2: Calculate First Difference

Determine the difference between the first and the second term: \(\frac{1}{3} - \frac{1}{2} = \frac{2}{6} - \frac{3}{6} = -\frac{1}{6}\).

Step 3: Calculate Second Difference

Determine the difference between the second and the third term: \(\frac{1}{4} - \frac{1}{3} = \frac{3}{12} - \frac{4}{12} = -\frac{1}{12}\).

Step 4: Calculate Third Difference

Determine the difference between the third and the fourth term: \(\frac{1}{5} - \frac{1}{4} = \frac{4}{20} - \frac{5}{20} = -\frac{1}{20}\).

Step 5: Compare the Differences

Compare the calculated differences: \(-\frac{1}{6}, -\frac{1}{12}, -\frac{1}{20}\). Since these differences are not the same, the sequence is not arithmetic.

Key Concepts

Common Difference

In an arithmetic sequence, the term "common difference" is crucial. It defines how each term in the sequence progresses from the one before it. This difference, denoted as \(d\), remains constant between any two consecutive terms.
Understanding how common differences work helps determine whether a sequence is arithmetic or not.

• Calculate the difference between each pair of consecutive terms.
• If all the differences are the same, the sequence is arithmetic.
• The consistent value is known as the common difference \(d\).

In the given sequence, we found differences between the terms \(\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}\), which were not constant. This inconsistency confirms that the sequence is not arithmetic, as there was no single common difference present across the terms.

Consecutive Terms

Consecutive terms in a sequence are terms that directly follow one another without any skips. In an arithmetic sequence, the relationship between any two consecutive terms is governed by the common difference. For a sequence to qualify as arithmetic, each consecutive term should be obtained by adding or subtracting the same value (the common difference) from the previous term.
When examining consecutive terms:

• First, list the terms in order.
• Then, calculate the differences to recognize any patterns.

From the original exercise, the sequence terms \(\frac{1}{2}, \frac{1}{3}\) are followed by \(\frac{1}{4}, \frac{1}{5}\). Checking their differences helps in identifying whether these terms maintain a logical and consistent progression necessary for an arithmetic sequence.

Sequence Differences

Sequence differences are the computed values obtained by subtracting one term in a sequence from the following term. These differences help identify the nature of the sequence. In arithmetic sequences, the sequence differences remain unchanged and constant; however, that was not the case in the provided exercise.
Let's examine why they are essential:

• Sequence differences reveal if the sequence is arithmetic.
• They aid in confirming whether a pattern or rule exists in the sequence.

In the given problem: we calculate \(\frac{1}{3} - \frac{1}{2} = -\frac{1}{6}\), \(\frac{1}{4} - \frac{1}{3} = -\frac{1}{12}\), and \(\frac{1}{5} - \frac{1}{4} = -\frac{1}{20}\). These differences are not consistent, indicating that the sequence does not have a uniform pattern typical of arithmetic sequences.