In mathematics, a **harmonic progression**(H.P.) is a progression formed by taking the reciprocals of an arithmetic progression.

The sequence is a harmonic progression when each term is the **harmonic mean of the neighboring terms**.

It is an infinite sequence of the form

\(\displaystyle {\frac {1}{a}},\ {\frac {1}{a+d}}\ ,{\frac {1}{a+2d}}\ ,{\frac {1}{a+3d}}\ ,\cdots\)where ‘*a’* is non zero and ‘*−a/d*‘ is not a natural number, or a finite sequence of the form

\(\displaystyle {\frac {1}{a}},\ {\frac {1}{a+d}}\ ,{\frac {1}{a+2d}}\ ,{\frac {1}{a+3d}}\ ,\cdots ,{\frac {1}{a+kd}} \)

where ‘*a’* is non zero, ‘*k’* is a natural number and ‘-a/d’ is not a natural number or is greater than ‘*k’*.

Index

**History**

The study of **harmonic sequences** dates to at least the *6th-century BCE* when the Greek philosopher and mathematician **Pythagoras** and his followers sought to explain through numbers the nature of the universe. One of the areas in which numbers were applied by the *Pythagoreans* was the *study of music*.

The seventeenth-century witnessed the classification of series into specific forms. In 1671 **James Gregory** used the term **infinite series** in connection with the infinite sequence. It was only through the rigorous development of *algebraic* and *set-theoretic tools* that the concepts related to sequence and series could be formulated suitably.

**Terms in a Harmonic Progression**

A sequence \(\frac{1}{a}, \frac{1}{a_1}, \frac{1}{a_2}, \frac{1}{a_3},…, \frac{1}{a_n},… \) is called an arithmetic sequence or arithmetic progression.

Here,

\(\frac{1}{a}\) is the **first term**.

\(\frac{1}{a_1}\) is the** second term**,

\(a_1\) can be given by \(a_1 = a + d\), and by taking reciprocal we can get \(\frac{1}{a_1} = \frac{1}{a + d}\).

\(\frac{1}{a_2}\) is the **third term**,

with \(a_2 = a_1 + d\) and taking reciprocal of this gives us \(\frac{1}{a_1 + d}\)

‘d’ is the **common difference** between two consecutive terms and it remains the same throughout a particular series.

Here we can get the generalized formula for the arithmetic sequence as:

\(l = \frac{1}{a + (n-1)d}\).

Here, \(a \neq 0\)

Where,

‘l’ is the** last term**

‘n’ is the **no. of term**

‘a’ is the **first term**

‘d’ is the **common difference**

**Sum of an Harmonic Progression**

**Infinite harmonic progressions are not summable.**

A harmonic progression of distinct unit fractions (other than the trivial case where a = 1 and d = 0) can’t sum to an integer. The reason is that at least one denominator of the progression will be divisible by a prime number that does not divide any other denominator.

**Application of a Harmonic Progression**

**Use in Geometry:** If collinear points A, B, C, and D are such that D is the harmonic conjugate (the two points that divide a line segment internally and externally in the same ratio) of C with respect to A and B, then the distances from any one of these points to the three remaining points form a harmonic progression.

In a **triangle**, if the altitudes are in arithmetic progression, then the sides are in harmonic progression.

**Leaning Tower of Lire:** In the Leaning Tower of Lire, uniform blocks are stacked on top of each other to achieve the maximum sideways or lateral distance covered. The blocks are stacked 1/2, 1/4,1/6, 1/8, 1/10… distance sideways below the original block. This ensures that the center of gravity is just at the center of the structure so that it does not collapse. A slight increase in weight on the structure causes it to become unstable and fall.

**Questions**

**Question 1.** Find the 4th and 8th term of the series 6, 4, 3, …

**Solution.** By reciprocal, we can turn this H.P. to an A.P.

\(\frac{1}{6},\frac{1}{4},\frac{1}{3}, …\)

Here, \(a_2 – a_1 = a_3 – a_2 = \frac{1}{12}\)

Therefore, \(\frac{1}{6},\frac{1}{4},\frac{1}{3}\) are in A.P.

4th term of this A.P. = \(\frac{1}{6} + 3 *\frac{1}{21} = \frac{1}{6} + \frac{1}{4} = \frac{5}{12}\)

8th term = \(\frac{1}{6} + 7 * \frac{1}{12} = \frac{9}{12}\).

Hence the 8th term of the H.P. is \(\frac{12}{9} = \frac{4}{3}\)

and the 4th term is \(\frac{12}{5}\).

**Question 2.** If the sixth term of an H.P. is 10 and the 11th term is 18. Find the 16th term.

**Solution.**

Here, \(a_6 = a + 5d = \frac{1}{10}\)

And \( a_{11} = a + 10d = \frac{1}{18}\)

Therefore, \(\frac{1}{10} + 5d = \frac{1}{18}\)

=> \(d = \frac{-2}{225}\)

=> \(a = \frac{13}{90}\)

Hence, \(a_{16} = a + 15d\)

=> \(a_{16} = \frac{13}{90} – \frac{2}{15}\)

=> \(a_{16} = \frac{1}{90}\)

Hence, the 16^{th} term will be 90.

**FAQs**

**What’s the difference between A.P. and H.P.**H.P. is \(\frac{1}{\mbox{A.P.}}\)

(OR) in simpler terms

if A.P. = a, b, c, d, e …

then, H.P. = \(\frac{1}{a}, \frac{1}{b}, \frac{1}{c}, \frac{1}{d}, \frac{1}{e} …\)

**What is the condition required to say that a sequence is in H.P.?**H.P. series when the reciprocals of elements are in A.P.

**How can we express the Harmonic Mean?**The general formula for calculating a harmonic mean is:

H.M. = \(\frac{n}{(\sum \frac{1}{x_i})}\),

where n is the no. of values and \(x_i\) is the term.

**Why is it called the harmonic series?**Its name comes from the concept of overtones, or harmonics in music: the wavelengths of the overtones of a vibrating string are 12, 13, 14, etc., of the string’s fundamental wavelength.