A series of numbers is said to be in harmonic sequence if the reciprocals of all the elements of the sequence form an arithmetic sequence.Its also called Harmonic Progression and denoted as H.P Here will teach you about Geometric Sequences and Series.

H.P = 1a, 1a + d, 1a+2d, 1a+(n-1)d, ….,1a+(n-1)d,….

Example: The sequence 1,2,3,4…. is an arithmetic progression, so its reciprocals 11, 12, 13, 14….. are a harmonic progression.

H.P = 1A.P

Theorem: A sequence is a harmonic progression if and only if its terms are the reciprocals of an arithmetic progression that doesn’t contain 0.0.

Harmonic Mean

The harmonic mean can be expressed as the reciprocal of the arithmetic mean of the reciprocals of the given set of observations.

If H(a,b) is the harmonic mean, A(a,b) is arithmetic mean and G(a,b) is geometric mean of real no. a and b.

A(a,b) = a + b2
G(a,b) = √ab

H(a,b) = 1A(1/a,1/b)
H(a,b) = 1a+b2ab
H(a,b) = 2aba + b

Relationship between H.P, A.P and G.P

H(a,b) = 2aba + b = aba+b2

H(a,b) = G(a,b)²A(a,b)

H(a,b).A(a,b) = G(a,b)²

I hope, this article will help you a lot to understand the Harmonic Sequence . If you still have any doubts and problems with any topic of mathematics you can ask your problem in the Ask Question section. You will get a reply shortly.

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