Harmonic Progression (HP), or Harmonic Sequence, is defined as a sequence of real numbers obtained by taking the reciprocals of an Arithmetic Progression that excludes 0. In this progression, each term is calculated as the harmonic mean of its two adjacent terms.
If the Arithmetic Progression is represented by the terms a, a + d, a + 2d, a + 3d, and so on, then the corresponding terms in the Harmonic Progression (or Harmonic sequence) are 1/a, 1/(a + d), 1/(a + 2d), 1/(a + 3d), 1/(a + 4d), and so forth up to 1/(a + (n - 1)d).
There are infinitely many examples of harmonic progression.
- Harmonic Progression of Natural Numbers: 1/1, 1/2, 1/3, 1/4 . . ..
- Leaning Tower of Lire: 1/2, 1/4, 1/6, 1/8, 1/10,...
- 1/3, 1/5, 1/7, 1/9 . . .
- 1/4, 1/8, 1/12, 1/16 . . .
- 1/5, 1/10, 1/15, 1/20 . . .
Formula for nth Term
When expressing the arithmetic progression in the format a, a + d, a + 2d, a + 3d, ..., a + (n − 1)d, the formula for the harmonic progression can be stated as follows: 1/a, 1/(a + d), 1/(a + 2d), 1/(a + 3d), and so on. The initial term is denoted by 'a' and the common difference by 'd'.
The general term (an) or nth term of the Harmonic Progression (H.P.) is given by the formula:
an = 1 / [a + (n - 1)d]
Where
- "a" represents the initial term of the Harmonic Progression (H.P.)
- "d" stands for the common difference between successive terms,
- "n" denotes the total number of terms in the Harmonic Progression (H.P.)
Sum of n Terms
Creating a harmonic progression, or 1/AP, is a straightforward process. Using the formula for the nth term in an arithmetic progression, a + (n-1)d, we can quickly generate the harmonic progression sequence. However, calculating the sum of this progression can be tedious. One approach involves either iterating through the sequence or employing approximations to create a formula that provides an accurate value up to a few decimal places.
To find the sum of n terms in a harmonic progression (Sn) for the sequence 1/a, 1/a + d, 1/a + 2d, ..., 1/a + (n−1)d, the formula is:
\text{S}_n \approx \frac{1}{d} \cdot \ln{\left( \frac{2a + (2n - 1)d}{2a -d}\right)} where,
- "a" denotes the first term of the Harmonic Progression (H.P.)
- "d" represents the common difference in the Harmonic Progression (H.P.)
- "ln" stands for the natural logarithm.
Note: This formula only gives approximate required value.
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Solved Examples on Harmonic Progression
Example 1: Find the value of the 21st term and the nth term of the harmonic progression 1/5, 1/9, 1/13, 1/14. . . .
Solution:
Given the Harmonic sequence: 1/5, 1/9, 1/13, 1/14, ...
Comparing this with the general Harmonic sequence formula 1/a, 1/(a + d), 1/(a + 2d), ... leads to the identification of the initial terms: 1/a = 1/5 and 1/(a + d) = 1/9.
By solving for 'a' and 'd', we find that a = 5 and d = 4.
To determine the 21st term, we utilize the formula for the nth term in a Harmonic sequence: 1/(a + (n - 1)d).
The 21st term = 1/(a + 20d) = 1/(5 + 20 x 4) = 1/(5 + 80) = 1/85.
The nth term = 1/(a + (n - 1)d) = 1/(5 + (n - 1)4) = 1/(5 + 4n - 4) = 1/(4n + 1)
Therefore, the 21st term is 1/85, and the nth term is 1/(4n + 1).
Example 2: Determine the 12th term of the harmonic progression if the fifth term is 1/16 and the eighth term is 1/25.
Solution:
Given;
The Fifth term = 1/(a + 4d) = 1/16
The Eight term = 1/(a + 7d) = 1/25
Comparing this with the general Harmonic sequence formula we have a + 4d = 16, and a + 7d = 25.
By solving for 'a' and 'd', we find that a = 4 and d = 3.
The 12th term of the Harmonic Progression = 1/(a + 11d) = 1/(4 + 11x3) = 1/(4 + 33) = 1/37.
Therefore, the 12th term of the harmonic progression is 1/37.
Example 3: Calculate the 16th term of the harmonic progression if the 6th and 11th terms of the harmonic progression are 10 and 18, respectively.
Solution:
For harmonic progression H is written in terms of A.P, we equate the sixth term of the A.P. to a + 5d = 1/10
And the eleventh term to a + 10d = 1/18
Solving these equations we get, a =13/90, and d = -2/ 225
To determine the 16th term, we use the expression
a + 15d = (13/90) – (2/15) = 1/90Thus the 16th term of the Harmonic Progression equates to the 1/16th term of the Arithmetic Progression (A.P.), which equals 90.