Harmonic progression (H.P.):

The sequence a₁, a₂, a₃.......a_n......(a_i ≠ 0) is known as an H.P. if the sequence is an A.P.

n^th Term of  H. P.:

The nth term, a_n, of the H.P. is  

Note

• There is no mathematical formula for the sum of n terms of an H.P.

Harmonic Means

• If a and b are two non-zero values, then the harmonic mean of b and a is b number H such that the numbers b, H, b are in H.P. We get

• If a₁, a₂, .......a_n are 'n' non-zero values, then the harmonic mean H of those numbers is provided by   .
• The n numbers H₁, H₂,.......,H_n are called n-harmonic means between a and b, if a , H₁ , H₂ ........, H_n , b are in H.P. i.e if  are in A.P. Consider d be the common difference of the A.P., then   

Problem: Calculate the 4th and the 8th terms of the H.P. 6, 4, 3,..........

Solution:              Consider 1/6,1/4,1/3,.........

                              Here         T₂ - T₁ = T₃ - T₂ = 1/12 => 1/6,1/4,1/3,..... is an A.P.

                              4th term of this A.P. = 1/6 + 3 x 1/12 = 1/6 + 1/4 = 5/12,

                              and the 8th term = 1/6 + 7 x 1/12 = 9/12

                              Hence the 4th term of the H.P. = 12/5 and the 8th term =12/9 = 4/3

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