What is Harmonic Progression in Mathematics?

What is Harmonic Progression in Mathematics?

Harmonic Progression (H.P.)

Definition:
A progression is called a harmonic progression (H.P.) if the reciprocals of its terms are in A.P.
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General term of an H.P.

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Harmonic mean

If three or more numbers are in H.P., then the numbers lying between the first and last are called harmonic means (H.M.’s) between them. For example 1, 1/3, 1/5, 1/7, 1/9 are in H.P. So 1/3, 1/5 and 1/7 are three H.M.’s between 1 and 1/9.
Also, if a, H, b are in H.P., then H is called harmonic mean between a and b.
(1) Insertion of harmonic means
(i) Single H.M. between a and b \(=\frac { 2ab }{ a+b }\).
(ii) H, H.M. of n non-zero numbers a1, a2, a3, ….. an is given by
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(iii) Let a, b be two given numbers. If n numbers H1, H2, H3, ….. Hn are inserted between a and b such that the sequence  a, H1, H2, H3, ….. Hn, b is a H.P., then H1, H2, H3, ….. Hn are called n harmonic means between a and b.
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Thus, if n harmonic means are inserted between two given numbers a and b, then the common difference of the corresponding A.P. is given by
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Properties of H.P.

  1. No term of H.P. can be zero.
  2. If H is the H.M. between a and b, then
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Harmonic Progression Problems with Solutions

1.
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2.
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Solution:
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3.
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4.
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5.
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6.
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7.
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