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.
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.