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Harmonic Progression

One can classify a harmonic progression (H.P.) as a sequence of real numbers which is determined by taking the reciprocals of the arithmetic progression (A.P.) that does not contain 0

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Posted by Kunduz Tutor, 27/10/2021
Harmonic Progression

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Harmonic Progression (H.P.) is a type of progression or sequence, which is the reciprocal form of the arithmetic progression (A.P.) with numbers that can never be 0. 

In other words, one can classify a harmonic progression (H.P.) as a sequence of real numbers which is determined by taking the reciprocals of the arithmetic progression (A.P.) that does not contain 0. Moreover, we shall be learning more about the harmonic progression, its formulas and properties.

Author – Ojasvi Chaplot


Introduction to Harmonic Progression

According to the definition, 

If, a sequence is given that A1, A2, A3, A4,….., An-1, An are in Harmonic Progression (H.P). Then, it can be said that 1/A1, 1/A2, 1/A3, 1/A4,….., 1/An-1, 1/An are in Arithmetic Progression (A.P). Harmonic Progression is also referred to as the H.P.

Important Trick

Whenever if an H.P. or a harmonic progression is given, then the first and foremost thing one must begin with is that always convert the harmonic progression into arithmetic progression.

After converting in arithmetic progression (A.P.), there comes the common difference. It can be stated that when the difference between two consecutive terms is always constant, then this constant is called as the common difference. Read here more about Arithmetic Progression.

For Example – If 1/2, 1/6, 1/10, 1/14, 1/18,….. are in harmonic progression, then, it can be concluded that 2, 6, 10, 14, 18,….. are in arithmetic progression with the value of common difference, d equal to 4.


Harmonic Progression Formula

According to the definition,

 If, a sequence is given that A1, A2, A3, A4,….., An-1, An are in Harmonic Progression (H.P).

Harmonic progression

Then, it can be said that 1/A1, 1/A2, 1/A3, 1/A4,….., 1/An-1, 1/An are in Arithmetic Progression (A.P).

Harmonic progression 1

Now, let us consider that a and b are respectively the first term and the second term of the harmonic progression, then its nth term will be:

A1 = a, A2 = b ———-(H.P.)

a1 = 1/a, a2 = 1/b ———–(A.P.)

So, the common difference, d = a2 – a1 

?d = 1/b – 1/a

? d = (a – b)/ab

The general term of A.P., an = a + (n – 1)d ; where a = first term and d = common difference.

? an = 1/a + (n – 1)(a – b)/ab

? an = [b + (n – 1)(a – b)]/ab ———-(in A.P.)

Therefore, the general term in harmonic progression (H.P.) can be written as An = 1/an

? An = (ab)/[b + (n – 1)(a – b)] ———-(in H.P.)

where, An is the general term of (H.P.).

Thus, the nth term of H.P. is equal to the reciprocal of the nth term of A.P.

So, Arithmetic Mean (A.M.) = (a1 + a2)/2

? A.M. = [1/a + 1/b]/2

? A.M. = (a + b)/2ab

Harmonic Mean (H.M.) = 1/A.M.

? H.M. = 2ab/(a + b)


Sum of Harmonic Progression(H.P)

If there are n number of terms in a harmonic sequence, then the sum of the arithmetic progression is calculated first by using the formula,

Sn = (n/2).[2a + (n − 1)d] or Sn = (n/2).[a + l]

Where, a = first term, d = common difference, l = last term in A.P.

Then, the sum of n terms in harmonic progression can be obtained by:

Sn’ = 1/Sn

i.e. Sum of n terms in H.P. is equal to the reciprocal of the Sum of n terms in A.P.

In particular, if we say that there is any specific formula for sum of H.P, then there is no such formula; and it can be obtained by converting H.P into arithmetic progression and then finding it.


Properties of Harmonic Progression

The properties of the geometric progression can be classified and listed as follows:

  • The first term and the common difference can be any real number.
  • A harmonic progression does not have common ratio.
  • In a H.P, any term can be expressed as harmonic mean (H.M.) of its equidistant terms, i.e. Ar = ( Ar-k + Ar+k)/2, where Ar is in H.P.

Illustrative Examples on Harmonic Progression

Example-1: Find the 10th term of a harmonic progression given as 1/3, 1/5, 1/7, 1/9,…..

Solution: 

Given that, 1/3, 1/5, 1/7, 1/9,….. ———-(in H.P.)

? 3, 5, 7, 9,….. ———-(in A.P.) with d = 2

So, the general term in A.P., 

? an = a + (n – 1)d

Now, 10th term in A.P. will be 

? a10 = 3 + (10 – 1)(2) ?a10 = 3 + 18

? a10 = 21.

So, the 10th term in H.P. is, A10 = 1/a10 = 1/21

Therefore, the 10th term of the H.P, A10 is 1/21.

Answer: The 10th term of the H.P, A10 = 1/21.


Example-2: Find the sum of the harmonic progression 1/6, 1/12, 1/18, 1/24.

Solution: 

Given that, 1/6, 1/12, 1/18, 1/24 ———-(in H.P.)

So, 6, 12, 18, 24 ———-(in A.P.) with common difference, d = 6

So, the Sum of A.P. is, 

? Sn = (n/2).[a + l] ? S4 = (4/2).[6 + 24] ? S4 = (2).[30]

? S4 = 60 ? S4’ = 1/S4 ? S4’ = 1/60

So, the sum of H.P. is, S4’ is equal to 1/60.

Answer: The sum of the H.P, S4’ = 1/60.


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