Problem 57 - Precalculus Mathematics for Calculus

Question

A sequence is harmonic if the reciprocals of the terms of the sequence form an arithmetic sequence. Determine whether the following sequence is harmonic: $$ 1, \frac{3}{5}, \frac{3}{7}, \frac{1}{3}, \ldots $$

Step-by-Step Solution

Answer

Yes, the sequence is harmonic.

Step 1: Definition of Harmonic Sequence

A sequence is harmonic if the reciprocals of its terms form an arithmetic sequence. That means if we have a sequence \(a_1, a_2, a_3, \ldots \), then \( \frac{1}{a_1}, \frac{1}{a_2}, \frac{1}{a_3}, \ldots \) should form an arithmetic sequence.

Step 2: Calculate Reciprocals

We calculate the reciprocals of the terms in the given sequence.1. The reciprocal of 1 is \(\frac{1}{1} = 1\).2. The reciprocal of \(\frac{3}{5}\) is \(\frac{5}{3}\).3. The reciprocal of \(\frac{3}{7}\) is \(\frac{7}{3}\).4. The reciprocal of \(\frac{1}{3}\) is 3.

Step 3: Check for Arithmetic Sequence in Reciprocals

To check if these reciprocals form an arithmetic sequence, the difference between consecutive terms must be the same.1. Calculate the differences: - Difference between \(1\) and \(\frac{5}{3}\): \(\frac{5}{3} - 1 = \frac{5}{3} - \frac{3}{3} = \frac{2}{3}\) - Difference between \(\frac{5}{3}\) and \(\frac{7}{3}\): \(\frac{7}{3} - \frac{5}{3} = \frac{2}{3}\) - Difference between \(\frac{7}{3}\) and 3: \(3 - \frac{7}{3} = \frac{9}{3} - \frac{7}{3} = \frac{2}{3}\)All differences are equal.

Step 4: Conclusion

Since the differences between consecutive reciprocals are constant, the sequence of reciprocals forms an arithmetic sequence. Therefore, the original sequence is harmonic.

Key Concepts

Arithmetic Sequence

Understanding an arithmetic sequence is key to identifying a harmonic sequence. An arithmetic sequence is a list of numbers where the difference between any two successive numbers is always the same. This difference is referred to as the 'common difference.' For instance, in the sequence 2, 4, 6, 8, ..., each term increases by 2. Here, 2 is the common difference. Similarity helps in solving problems where we need to determine a pattern or predict future numbers in a sequence. If the sequence of reciprocals (of an original sequence) is arithmetic, only then is the original termed harmonic.

Reciprocals

The concept of reciprocals is foundational when analyzing harmonic sequences. The reciprocal of a number is simply 1 divided by that number. For example, the reciprocal of 2 is \( \frac{1}{2} \), and the reciprocal of \( \frac{3}{5} \) is \( \frac{5}{3} \). When determining if a sequence is harmonic, we first find the reciprocals of each term. Once found, we then examine these reciprocals to check if they form an arithmetic sequence. Understanding how to transform each term to its reciprocal is crucial in this process.

Problem Solving

Approaching sequence problems systematically is essential. Begin by clearly defining the problem and the type of sequence. Here, we need to ascertain if a sequence is harmonic by checking if its reciprocals form an arithmetic sequence. Calculate each term's reciprocal, as shown in the exercise, and determine if these reciprocals form a pattern with a consistent common difference. This step-by-step methodology assures clarity and reduces errors. Always double-check calculations, especially when dealing with fractions and differences, to ensure the final conclusion is accurate.