Difference between revisions of "Harmonic sequence"
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=== Example 2 === | === Example 2 === | ||
− | ''Let <math>a</math>, <math>b</math>, and <math>c</math> be positive real numbers. Show that if <math>a, b, and c</math> are in harmonic progression, then <math>a/(b+c), b/(c+a), c(a+b)</math> | + | ''Let <math>a</math>, <math>b</math>, and <math>c</math> be positive real numbers. Show that if <math>a</math>, <math>b</math>, and <math>c</math> are in harmonic progression, then <math>a/(b+c)</math>, <math>b/(c+a)</math>, and <math>c/(a+b)</math> are as well.'' |
'''Solution''': Using the harmonic mean property of harmonic sequences, we are given that <math>1/a + 1/b = 2/c</math>, and we wish to show that <math>(b+c)/a + (a+b)/c = 2(c+a)/b</math>. We work backwards from the latter equation. | '''Solution''': Using the harmonic mean property of harmonic sequences, we are given that <math>1/a + 1/b = 2/c</math>, and we wish to show that <math>(b+c)/a + (a+b)/c = 2(c+a)/b</math>. We work backwards from the latter equation. |
Revision as of 20:43, 26 November 2021
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. By definition, can never be a term of a harmonic sequence.
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
[hide]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 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; this is by a limit comparison test with the series , which is referred to as the harmonic series. As for finite harmonic series, a general expression for the sum has ever been found. One must find a strategy to evaluate their sum on a case-by-case basis.
Examples
Here are some example solutions 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 yields Because , , and were all defined to be 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. Then , which implies that the sequence is in harmonic progression, 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, . By an above lemma, the entire sequence is in harmonic progression, which means that we can apply tools of harmonic sequences to this problem.
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 , or .
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.