harmonic sequence (harmonic progression)

Harmonic Sequence (Harmonic Progression) in Math

Definition

In mathematics, a harmonic sequence, also known as a harmonic progression, is a sequence of numbers in which the reciprocal of each term is in arithmetic progression. In simpler terms, a harmonic sequence is a sequence of numbers where the difference between consecutive terms is a constant reciprocal.

Knowledge Points

A harmonic sequence contains the following key points:

1. Reciprocal: Each term in a harmonic sequence is the reciprocal of a corresponding term in an arithmetic sequence.
2. Arithmetic Progression: The reciprocals of the terms in a harmonic sequence form an arithmetic progression.
3. Constant Difference: The difference between consecutive terms in a harmonic sequence is constant.

Formula or Equation

The formula for the nth term of a harmonic sequence is given by:

where:

• a_n is the nth term of the harmonic sequence,
• d is the common difference between consecutive terms in the arithmetic progression of reciprocals,
• n is the position of the term in the sequence, and
• c is a constant term.

Application

To apply the formula for a harmonic sequence, you need to know the values of d, n, and c. By substituting these values into the formula, you can find the corresponding term in the sequence.

Symbol

There is no specific symbol for a harmonic sequence. It is usually denoted by the terms "harmonic sequence" or "harmonic progression."

Methods

There are several methods to work with harmonic sequences, including:

1. Finding the nth term using the formula.
2. Finding the sum of a given number of terms in a harmonic sequence.
3. Determining if a given sequence is a harmonic sequence or not.

Solved Examples

1. Find the 5th term of the harmonic sequence with a common difference of 2 and a constant term of 3.

Solution: Using the formula, we have: a_5 = 1 / (2 * 5 + 3) = 1/13

2. Determine if the sequence 1/2, 1/4, 1/6, 1/8 is a harmonic sequence.

Solution: The reciprocals of the terms form an arithmetic progression: 2, 4, 6, 8. Therefore, the given sequence is a harmonic sequence.

Practice Problems

1. Find the 10th term of the harmonic sequence with a common difference of 3 and a constant term of 2.
2. Calculate the sum of the first 6 terms of the harmonic sequence with a common difference of 1/2 and a constant term of 1.

FAQ

Q: What is a harmonic sequence (harmonic progression)? A: A harmonic sequence is a sequence of numbers in which the reciprocal of each term is in arithmetic progression.

Q: How do you find the nth term of a harmonic sequence? A: The nth term of a harmonic sequence can be found using the formula a_n = 1 / (d * n + c), where d is the common difference, n is the position of the term, and c is a constant term.

Q: How do you determine if a sequence is a harmonic sequence? A: To determine if a sequence is a harmonic sequence, check if the reciprocals of the terms form an arithmetic progression. If they do, the sequence is a harmonic sequence.