Progression also known as sequence or series, are the types of numbers which are placed in a particular order to form a recognizable table. Simply we can say that harmonic progression is the reciprocal of the values of the terms in arithmetic progression. It can be explained as if the terms of arithmetic progression like a, b, c, d, e, f are available in the form of 1/a, 1/b, 1/c, 1/d, 1/e, 1/f, in which terms of harmonic progression can be written as 1/a, 1/(a + d), 1/(a + 2d), etc. A sequence can be said in harmonic progression only when the terms are in harmonic mean with their neighboring terms.

Harmonic mean is to find out the average speed of the journey where the average formula of the speed can be given by 2ab/ (a + b) kmph, where a and b are the speed at which a person travels a distance of A and B. Harmonic mean can be easily calculated with the help of the following example;

If three terms a, b and c are in harmonic progression then, the terms in arithmetic progression include 1/a, 1/b, 1/c,

• 2/b = 1/a + 1/c

• b = 2ac/ (a + c)

In the above equation, b is the harmonic mean of its other two neighboring terms (a, c).

__Conditions to solve the terms of harmonic progression:__• Mostly all the terms included in the harmonic progression are first converted into arithmetic progression and then are solved further.

• To solve the problems in harmonic progression, one must make a corresponding arithmetic series in order to solve the harmonic series.

• For any two given terms, if A.P, G.P, H.P are indicated as arithmetic, geometry and harmonic progression, then

A .P≥ G.P ≥ H.P

And, (A.P) (H.P) = (G.P)^{2}• Through the given terms of arithmetic terms, we need to first find the first three terms of the harmonic progression and then they can be taken in its general form of 1/(a – d), 1/a, 1/(a + d).

• If the n^{th} term of the arithmetic progression is given by an = a + (n – 1) d, and we know to solve the terms in harmonic progression, we first have to convert the terms of A.P in H.P, so the n^{th} term of the harmonic progression can be given by 1/ [a + (n - 1) d].

• Any terms available in the series of harmonic progression cannot be zero.

__Arithmetic and geometric progression:__

d = a2 – a1 = a3 – a2 = a4 – a3 = ...

If one term and the common difference is known then we can easily calculate the value of the next terms by adding the value of the common difference in the value of the term given, which can be given by as follows;

A(m+1) = am + d

The value of the nth term in an arithmetic progression can be calculated easily with the help of the following formula;

an = a1 + (n − 1) d

or, an =am + (n − m) d

The sum of the value of n terms in an arithmetic progression can be given as the n times average of the sum of first and the last term of the arithmetic progression, which can be mathematically shown as follows;

S = (n/2) (a1 + an)

If the last term of the sequence (an) is not given but the value of common difference is known then the sum of the n terms can be calculated easily with the help of the following formulae;

S = (n/2) [2a1 + (n − 1) d]

Geometric progression are the terms in which if the first term is known and the common ratio is known then the next term of the series can be calculated easily by multiplying the first term with the fixed and non-zero term of common ratio which is denoted by r.

The common difference of any geometric progression can be easily calculated by dividing the second term with the first term and the quotient remaining is known as common difference, i.e.

r = (am+1 / am) = a2/ a1 = a3/ a2 = a4/ a3 = ...

If the value of the first term and the common difference is known then we can easily calculate the value of the next term or the nth term with the help of the following formulae:

an = a1 rn-1

or, an = am rn-m

The sum of all the terms included in the sequence of geometric progression can be easily calculated with the help of the common difference and the known value of the first term in the sequence, which can be easily calculated with the help of the following formulae;

S = [a1 (1 − rn)]/ (1 – r)

Relation between A.P, G.P, and H.P:-

• To find the values of three numbers in H.P can be easily calculated with the help of its general formulae, i.e. 1/(a – d), 1/a, 1/(a + d).

• The relation between the three types of mean available in the sequence can be provided as follows;

A.M > G.M > H.M.

Where A.M is the Arithmetic Mean, G.M is the Geometric Mean and the H.M is known as Harmonic Mean.

• If a and b are the two real positive numbers given then the relation between the three types of the sequence can be given by as follows;

A.M x H.M = G.M2

Where A.M is the Arithmetic Mean, G.M is the Geometric Mean and the H.M is known as Harmonic Mean.

• The nth term of the arithmetic progression can be easily calculated by an = a + (n - 1) d, while the nth term of the harmonic progression can be easily calculated by 1/ [a + (n - 1) d].

• The most basic relation between H.P and A.P is that most of the H.P terms are calculated by first converting them into the terms of A.P.

From the above information we can conclude that the nth term of the Harmonic Progression can be calculated with the following formulae;

The nth term of the Harmonic Progression = 1/ (nth term of the corresponding Arithmetic Progression).

To check whether the sequence is in harmonic progression or not we must check if the reciprocal of the given sequence has the same difference between their consecutive terms and can be called as an arithmetic sequence then the given sequence whose reciprocal was done can be said in harmonic progression.

Some examples are shown below to explain the concept of harmonic progression:-

Example 1

If the 7th and the 12th term of an H.P are given as 1/10 and 1/25 respectively, then to find the 20th and the nth term the following solution can be taken into consideration.

Solution:

The solution of the above example can be given with the help of the general form of the Harmonic progression.

The general form of Harmonic progression is;

1/a + 1/(a + d) + 1/(a + 2d) + ….

For the nth term, the general form is given as 1/ [a + (n – 1) d] and by putting the values of a and d from equation (3) in this form we can easily calculate the value of the nth term of the harmonic sequence.

nth term = 1/ [a + (n – 1) d]

nth term = 1/ [-8 + (n – 1) 3]

nth term = 1/ (3n – 11)

Hence, the values of 20th and the nth term are easily found out in this above example with the help of the two terms of the harmonic progression provided.

Harmonic mean is to find out the average speed of the journey where the average formula of the speed can be given by 2ab/ (a + b) kmph, where a and b are the speed at which a person travels a distance of A and B. Harmonic mean can be easily calculated with the help of the following example;

If three terms a, b and c are in harmonic progression then, the terms in arithmetic progression include 1/a, 1/b, 1/c,

In the above equation, b is the harmonic mean of its other two neighboring terms (a, c).

A .P≥ G.P ≥ H.P

And, (A.P) (H.P) = (G.P)

d = a2 – a1 = a3 – a2 = a4 – a3 = ...

If one term and the common difference is known then we can easily calculate the value of the next terms by adding the value of the common difference in the value of the term given, which can be given by as follows;

A(m+1) = am + d

The value of the nth term in an arithmetic progression can be calculated easily with the help of the following formula;

an = a1 + (n − 1) d

or, an =am + (n − m) d

The sum of the value of n terms in an arithmetic progression can be given as the n times average of the sum of first and the last term of the arithmetic progression, which can be mathematically shown as follows;

S = (n/2) (a1 + an)

If the last term of the sequence (an) is not given but the value of common difference is known then the sum of the n terms can be calculated easily with the help of the following formulae;

S = (n/2) [2a1 + (n − 1) d]

Geometric progression are the terms in which if the first term is known and the common ratio is known then the next term of the series can be calculated easily by multiplying the first term with the fixed and non-zero term of common ratio which is denoted by r.

The common difference of any geometric progression can be easily calculated by dividing the second term with the first term and the quotient remaining is known as common difference, i.e.

r = (am+1 / am) = a2/ a1 = a3/ a2 = a4/ a3 = ...

If the value of the first term and the common difference is known then we can easily calculate the value of the next term or the nth term with the help of the following formulae:

an = a1 rn-1

or, an = am rn-m

The sum of all the terms included in the sequence of geometric progression can be easily calculated with the help of the common difference and the known value of the first term in the sequence, which can be easily calculated with the help of the following formulae;

S = [a1 (1 − rn)]/ (1 – r)

Relation between A.P, G.P, and H.P:-

The above shown figure easily explains the relation between the different types of sequences and can be explained further in details:

A.M > G.M > H.M.

Where A.M is the Arithmetic Mean, G.M is the Geometric Mean and the H.M is known as Harmonic Mean.

A.M x H.M = G.M2

Where A.M is the Arithmetic Mean, G.M is the Geometric Mean and the H.M is known as Harmonic Mean.

From the above information we can conclude that the nth term of the Harmonic Progression can be calculated with the following formulae;

The nth term of the Harmonic Progression = 1/ (nth term of the corresponding Arithmetic Progression).

To check whether the sequence is in harmonic progression or not we must check if the reciprocal of the given sequence has the same difference between their consecutive terms and can be called as an arithmetic sequence then the given sequence whose reciprocal was done can be said in harmonic progression.

Some examples are shown below to explain the concept of harmonic progression:-

Example 1

If the 7th and the 12th term of an H.P are given as 1/10 and 1/25 respectively, then to find the 20th and the nth term the following solution can be taken into consideration.

Solution:

The solution of the above example can be given with the help of the general form of the Harmonic progression.

The general form of Harmonic progression is;

1/a + 1/(a + d) + 1/(a + 2d) + ….

The value of the 7th term of the sequence is given as 1/10.

And according to the general formula, the 7th term can be given as 1/(a + 6d)

So by combining the above two equations for the 7th term, we get a + 6d = 10 …… (1)

And according to the general formula, the 7th term can be given as 1/(a + 6d)

So by combining the above two equations for the 7th term, we get a + 6d = 10 …… (1)

Similarly, for the 12th term

12th term = 1/25 and 1/(a + 11d)

So after combining we get a + 11d = 25 ...... (2)

From the above two equations ( 1 and 2 ) we get,

A = -8 and d = 3 ……. (3)

12th term = 1/25 and 1/(a + 11d)

So after combining we get a + 11d = 25 ...... (2)

From the above two equations ( 1 and 2 ) we get,

A = -8 and d = 3 ……. (3)

Through these values of a and d, we can easily calculate the values of 20th as well as the nth term of the harmonic progression.

For the 20th term, the general form is 1/ (a + 19d) and putting the values of equation (3) in this form we get,

20th term = 1/ [-8 + (19*3)]

20th term = 1/ 49

For the 20th term, the general form is 1/ (a + 19d) and putting the values of equation (3) in this form we get,

20th term = 1/ [-8 + (19*3)]

20th term = 1/ 49

Similarly, we can easily calculate the nth term of the harmonic progression with the help of its general form and the values of a and d given in equation (3).

For the nth term, the general form is given as 1/ [a + (n – 1) d] and by putting the values of a and d from equation (3) in this form we can easily calculate the value of the nth term of the harmonic sequence.

nth term = 1/ [a + (n – 1) d]

nth term = 1/ [-8 + (n – 1) 3]

nth term = 1/ (3n – 11)

Hence, the values of 20th and the nth term are easily found out in this above example with the help of the two terms of the harmonic progression provided.