Harmonic sequence
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 .
Contents
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1 Properties
2 Sum
3 Examples
o 3.1 Example 1
o 3.2 Example 2
o 3.3 Example 3
4 More Problems
o 4.1 Introductory
5 See Also
Properties
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.
Sum
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.
Examples
Here are some example problems that utilize harmonic sequences and series.
Example 1
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.
Example 2
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.
Example 3
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 TOTO SLOT 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 .
More Problems
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 .
