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.
A harmonic sequence contains the following key points:
The formula for the nth term of a harmonic sequence is given by:
where:
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.
There is no specific symbol for a harmonic sequence. It is usually denoted by the terms "harmonic sequence" or "harmonic progression."
There are several methods to work with harmonic sequences, including:
Solution: Using the formula, we have: a_5 = 1 / (2 * 5 + 3) = 1/13
Solution: The reciprocals of the terms form an arithmetic progression: 2, 4, 6, 8. Therefore, the given sequence is a harmonic sequence.
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.