Problem 69 - 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: $$1, \frac{3}{5}, \frac{3}{7}, \frac{1}{3}, \dots$$

Step-by-Step Solution

Answer

Yes, the sequence is harmonic.

Step 1: Understand the Definition

A sequence is harmonic if the reciprocals of its terms form an arithmetic sequence. An arithmetic sequence has a constant difference between consecutive terms.

Step 2: Find Reciprocals of the Sequence Terms

Take the given sequence: \(1, \frac{3}{5}, \frac{3}{7}, \frac{1}{3}\) and find the reciprocals: \(1, \frac{5}{3}, \frac{7}{3}, 3\).

Step 3: Check for Arithmetic Sequence

Calculate the differences between consecutive reciprocals: \( \frac{5}{3} - 1 = \frac{2}{3} \), \( \frac{7}{3} - \frac{5}{3} = \frac{2}{3} \), \( 3 - \frac{7}{3} = \frac{2}{3} \). All differences are equal.

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is known as the "common difference." For example, in the sequence 2, 4, 6, 8, the common difference is 2. To find the common difference, subtract any term from the term that comes after it.

• Formula: For a sequence \( a_1, a_2, a_3, \dots \), the common difference \( d = a_{n+1} - a_n \).
• Example: In the sequence 5, 10, 15, 20, \( d = 10 - 5 = 5 \).
• Properties: The sequence continues infinitely with the same difference between each term.

In the context of the harmonic sequence, the reciprocals of the terms must form an arithmetic sequence to define the original sequence as harmonic. This means checking for equal differences between consecutive reciprocal terms should yield a constant value, confirming the harmonic nature.

Reciprocal

In mathematics, a reciprocal of a number is one divided by that number. The reciprocal flips the numerator and denominator of a fraction. For instance, the reciprocal of \( \frac{3}{5} \) is \( \frac{5}{3} \).

• Definition: For a number \( a \), the reciprocal is \( \frac{1}{a} \).
• Use: The reciprocal is used to turn division into multiplication. For example, dividing by a number is the same as multiplying by its reciprocal.
• Important Note: The reciprocal of 1 is 1, as \( \frac{1}{1} = 1 \), and the reciprocal of 0 is undefined.

In the given exercise, finding the reciprocals of the sequence's terms was crucial. These reciprocals are then analyzed to check if they form an arithmetic sequence, which determines if the original sequence is harmonic.

Sequence Analysis

Sequence analysis involves examining the properties and behavior of sequences, particularly identifying patterns or rules. In this exercise, we analyzed a sequence by evaluating its reciprocal terms and assessing whether they form an arithmetic sequence.

Steps in Analyzing a Sequence:

• Identify the type of sequence: Determine if it is arithmetic, geometric, harmonic, etc.
• Calculate relevant mathematic properties: For arithmetic sequences, find the common difference; for geometric, the common ratio.
• Examine changes between terms: This helps in confirming a pattern or rule governing the sequence.

For this exercise, the reciprocal sequence: \(1, \frac{5}{3}, \frac{7}{3}, 3\) showed constant differences between consecutive terms \(\frac{2}{3}\). Hence, confirming these reciprocals form an arithmetic sequence, and thereby verifying the original sequence as harmonic. Understanding and applying these analytical steps is key to mastering sequence-related problems.