Sequence and Series-4

Harmonic Progression (H.P.)

Definition:

·   A progression is called a harmonic progression (H.P.) if the reciprocals of its terms are in A.P.

Standard form: 1/a, 1/(a +d), 1/(a +2d)............  1, 1/2, 1/3,..............are in harmonic progression and 1, 2, 3.....are in arithmetic progression.

General term of an H.P.

harmonic progression

 

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.

 

Insertion of Harmonic Means: 

Single H.M. between a and b = 2ab/(a +b)

H, H.M. of n non-zero numbers a1, a2, a3, a4,......an   is given by ;

harmonic means

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.....Hare called n harmonic means between a and b.

Now, H1, H2, H3.....Hn  b are in H.P. 

1/a,  1/H1, 1.H2, 1/H3.....1/Hn,  1/b are in AP.

Let D be the common difference of this A.P. Then,

harmonic series means.png

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

 harmonic sequence

 

Properties of H.P. :

No terms of HP can be zero

 If H is the H.M. between a and b, then

 

progressions