In algebra, a harmonic sequence, sometimes called a harmonic progression, is a sequence of numbers such that the difference between the reciprocals of any two consecutive terms is constant. In other words, a harmonic sequence is formed by taking the reciprocals of every term in an arithmetic sequence.
For example, and are harmonic sequences; however, and are not.
More formally, a harmonic progression biconditionally satisfies A similar definition holds for infinite harmonic sequences. It appears most frequently in its three-term form: namely, that constants , , and are in harmonic progression if and only if .
Because the reciprocals of the terms in a harmonic sequence are in arithmetic progression, one can apply properties of arithmetic sequences to derive a general form for harmonic sequences. Namely, for some constants and , the terms of any finite harmonic sequence can be written as
A common lemma is that a sequence is in harmonic progression if and only if is the harmonic mean of and for any consecutive terms . In symbols, . This is mostly used to perform substitutions, though it occasionally serves as a definition of harmonic sequences.
A harmonic series is the sum of all the terms in a harmonic series. All infinite harmonic series diverges, which follows by the limit comparison test with the series . This series is referred to as the harmonic series. As for finite harmonic series, there is no known general expression for their sum; one must find a strategy to evaluate one on a case-by-case basis.
Here are some example problems that utilize harmonic sequences and series.
Find all real numbers such that is a harmonic sequence.
Solution: Using the harmonic mean properties of harmonic sequences, Note that would create a term of —something that breaks the definition of harmonic sequences—which eliminates them as possible solutions. We can thus multiply both sides by to get . Expanding these factors yields , which simplifies to . Thus, is the only solution to the equation, as desired.
Let , , and be positive real numbers. Show that if , , and are in harmonic progression, then , , and are as well.
Solution: Using the harmonic mean property of harmonic sequences, we are given that , and we wish to show that . We work backwards from the latter equation.
One approach might be to add to both sides of the equation, which when combined with the fractions returns Because , , and are all positive, . Thus, we can divide both sides of the equation by to get , which was given as true.
From here, it is easy to write the proof forwards. Doing so proves that , which implies that , , is a harmonic sequence, as required.
2019 AMC 10A Problem 15: A sequence of numbers is defined recursively by , , and for all Then can be written as , where and are relatively prime positive integers. What is ?
Solution: We simplify the series' recursive formula. Taking the reciprocals of both sides, we get the equality Thus, . This is the harmonic mean, which implies that is a harmonic progression. Thus, the entire sequence is in harmonic progression.
Using the tools of harmonic sequences, we will now find a closed expression for the sequence. Let and . Simplifying the first equation yields and substituting this into the second equation yields . Thus, and so . The answer is then .
Here are some more problems that utilize harmonic sequences and series. Note that harmonic sequences are rather uncommon compared to their arithmetic and geometric counterparts.