Harmonic Progression

In Mathematics, a progression is defined as a series of numbers arranged in a predictable pattern. It is a type of number set which follows specific, definite rules. There is a difference between the progression and a sequence. A progression has a particular formula to compute its nth term, whereas a sequence is based on the specific logical rules. A progression can be generally classified into three different types, such as Arithmetic Progression, Geometric Progression and Harmonic Progression. In this article, we are going to discuss the harmonic progression sum formula with its examples.

Table of Contents:

• Definition
• Formula
• Harmonic Progression Sum
• Relation Between AP, GP and HP
• Examples

What is Harmonic Progression?

A Harmonic Progression (HP) is defined as a sequence of real numbers which is determined by taking the reciprocals of the arithmetic progression that does not contain 0. In harmonic progression, any term in the sequence is considered as the harmonic means of its two neighbours. For example, the sequence a, b, c, d, …is considered as an arithmetic progression; the harmonic progression can be written as 1/a, 1/b, 1/c, 1/d, …

Harmonic Mean: Harmonic mean is calculated as the reciprocal of the arithmetic mean of the reciprocals. The formula to calculate the harmonic mean is given by:

Harmonic Mean = n /[(1/a) + (1/b)+ (1/c)+(1/d)+….]

Where,

a, b, c, d are the values and n is the number of values present.

Harmonic Progression Formula

To solve the harmonic progression problems, we should find the corresponding arithmetic progression sum. It means that the nth term of the harmonic progression is equal to the reciprocal of the nth term of the corresponding A.P. Thus, the formula to find the nth term of the harmonic progression series is given as:

The nth term of the Harmonic Progression (H.P) = 1/ [a+(n-1)d]

Where

“a” is the first term of A.P

“d” is the common difference

“n” is the number of terms in A.P

The above formula can also be written as:

The nth term of H.P = 1/ (nth term of the corresponding A.P)

Harmonic Progression Sum

If 1/a, 1/a+d, 1/a+2d, …., 1/a+(n-1)d is given harmonic progression, the formula to find the sum of n terms in the harmonic progression is given by the formula:

Sum of n terms,

Where,

“a” is the first term of A.P

“d” is the common difference of A.P

“ln” is the natural logarithm

Relation Between AP, GP and HP

For any two numbers, if A.M, G.M, H.M are the Arithmetic, Geometric and Harmonic Mean respectively, then the relationship between these three is given by:

• G.M² = A.M × H.M, where A.M, G.M, H.M are in G.P
• A.M ≥ G.M ≥ H.M

Harmonic Progression Examples

Here, solved problems on the harmonic progression are given.

Example 1:

Determine the 4th and 8th term of the harmonic progression 6, 4, 3,…

Solution:

Given:

H.P = 6, 4, 3

Now, let us take the arithmetic progression from the given H.P

A.P = ⅙, ¼, ⅓, ….

Here, T₂ -T_{1 =} T₃ -T₂ = 1/12 = d

So, in order to find the 4th term of an A. P, use the formula,

The nth term of an A.P = a+(n-1)d

Here, a = ⅙, d= 1/12

Now, we have to find the 4th term.

So, take n=4

Now put the values in the formula.

4th term of an A.P = (⅙) +(4-1)(1/12)

= (⅙)+(3/12)

= (⅙)+ (¼)

= 5/12

Similarly,

8th term of an A.P = (⅙) +(8-1)(1/12)

= (⅙)+(7/12)

= 9/12

Since H.P is the reciprocal of an A.P, we can write the values as:

4th term of an H.P = 1/4th term of an A.P = 12/5

8th term of an H.P = 1/8th term of an A.P = 12/9 = 4/3

Example 2:

Compute the 16th term of HP if the 6th and 11th term of HP are 10 and 18, respectively.

Solution:

The H.P is written in terms of A.P are given below:

6th term of A.P = a+5d = 1/10 —- (!)

11th term of A.P = a+10d = 1/18 ……(2)

By solving these two equations, we get

a =13/90, and d = -2/ 225

To find 16th term, we can write the expression in the form,

a+15d = (13/90) – (2/15) = 1/90

Thus, the 16th term of an H.P = 1/16th term of an A.P = 90

Therefore, the 16th term of the H.P is 90.

Practice Problems on Harmonic Progression

Solve the harmonic progressions practice problems provided below:

1. The second and the fifth term of the harmonic progression is 3/14 and 1/10. Compute the sum of 6th and 7th term of the series.
2. The sum of the reciprocals of the first 11 terms in the harmonic progression series is 110. Determine the 6 terms of the harmonic progression series.

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