Problem 53 - Algebra and Trigonometry

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

The sequence is harmonic, as the reciprocals form an arithmetic sequence.

Step 1: Define Harmonic Sequence

A sequence is defined as harmonic if the reciprocals of its terms form an arithmetic sequence. In an arithmetic sequence, the difference between any two consecutive terms is constant.

Step 2: Find the Reciprocals of the Given Sequence

Find the reciprocals of each term in the sequence \(1, \frac{3}{5}, \frac{3}{7}, \frac{1}{3}\ldots\). The reciprocals are: \(1, \frac{5}{3}, \frac{7}{3}, 3, \ldots\).

Step 3: Check if Reciprocals Form an Arithmetic Sequence

To determine if the reciprocals \(1, \frac{5}{3}, \frac{7}{3}, 3\) are in an arithmetic sequence, find the differences between consecutive terms: \(\frac{5}{3} - 1 = \frac{2}{3}\), \(\frac{7}{3} - \frac{5}{3} = \frac{2}{3}\), and \(3 - \frac{7}{3} = \frac{2}{3}\). Each difference is \(\frac{2}{3}\), a constant value, indicating that the reciprocals form an arithmetic sequence.

Step 4: Conclusion

Since the reciprocals of the terms form an arithmetic sequence, the given sequence \(1, \frac{3}{5}, \frac{3}{7}, \frac{1}{3}, \ldots\) is a harmonic sequence.

Key Concepts

Reciprocal

In mathematics, the reciprocal of a number is simply 1 divided by that number. It is also known as the multiplicative inverse. For example, the reciprocal of 2 is \( \frac{1}{2} \), and the reciprocal of \( \frac{3}{4} \) is \( \frac{4}{3} \). Understanding reciprocals is essential when dealing with harmonic sequences since the definition of a harmonic sequence involves taking the reciprocal of each term.

Let's consider the sequence \( 1, \frac{3}{5}, \frac{3}{7}, \frac{1}{3}, \ldots \). By taking the reciprocal, we transform these terms into \( 1, \frac{5}{3}, \frac{7}{3}, 3, \ldots \). These new numbers are what we use to examine further properties, like forming another kind of sequence.

Reciprocals have some interesting properties:

• The reciprocal of a reciprocal brings you back to your original number.
• Multiplying a number by its reciprocal always equals 1.
• The concept of reciprocity is key in many mathematical procedures, such as dividing fractions.

Arithmetic Sequence

An arithmetic sequence, sometimes called an arithmetic progression, is a sequence of numbers in which the difference between consecutive terms is always the same. This consistent difference is known as the common difference. An example would be the sequence \( 3, 6, 9, 12, \ldots \) with a common difference of 3.

In our exercise, we took the reciprocals of the harmonic sequence's terms, which resulted in \( 1, \frac{5}{3}, \frac{7}{3}, 3, \ldots \). To identify whether these values form an arithmetic sequence, we look at the difference between each pair of consecutive terms:

• \( \frac{5}{3} \) minus \( 1 = \frac{2}{3} \)
• \( \frac{7}{3} \) minus \( \frac{5}{3} = \frac{2}{3} \)
• \( 3 \) minus \( \frac{7}{3} = \frac{2}{3} \)

All these differences are \( \frac{2}{3} \), confirming it's indeed an arithmetic sequence.

Sequence Analysis

Sequence analysis involves examining the properties and behavior of a sequence to understand its characteristics. In particular, sequence analysis is crucial when determining whether a sequence is harmonic.

In our case, after calculating the reciprocals of the terms in the provided sequence, we performed an analysis to check if these reciprocals form an arithmetic sequence. This type of sequence analysis reveals whether the initial sequence can be classified as harmonic. The importance of identifying harmonic sequences extends to various fields such as physics and engineering, where understanding the relationship between sequential elements is vital.

To conduct successful sequence analysis:

• Understand the definitions (like arithmetic or geometric sequences).
• Identify the type of sequence you have by looking for patterns and consistent relationships.
• Apply algebraic methods to find common traits such as a constant difference or ratio.

This organized approach helps in breaking down complex problems into simpler parts, making it easier to interpret the results and make informed conclusions.