Mathematical Series: Sum of Harmonic Progression

by Yuvi K - December 16, 2023

## What is Harmonic Progression?

Harmonic Progression (धारण प्र在ग्रेशन) is a type of mathematical sequence or progression where the reciprocals of the consecutive terms are in arithmetic progression. This type of progression is often used in acoustics for the study of notes and chords in musical harmony. This article discusses the sum of a harmonic progression of 500 to 800 terms in detail.

## Sum of Harmonic Progression

In mathematics, the sum of harmonic progression from 500 to 800 terms, (500, 8001) is an expression used to represent what is known as the harmonic series. The harmonic series is an infinite series that can be thought of as mathematical ‘patterns’ of numbers that continue infinitely. The sum of harmonic progression from 500 to 800 can be calculated by using the following formula:

Sum = ᵍⁿ ᶬᵐ – 1

The above formula is used to calculate the sum of the harmonic progression when we know the number of terms in the series.

In this case, ᵍ = the first term in the series or 500 and n is the number of terms in the series or 800.

Thus, the sum of harmonic progression of 500 to 800 is expressed as follows:

Sum = 500 ᶬᵐ800 – 1

Expanding this equation, we get:

Sum = (500/800)⁸⁰⁰ – 1

Now, if we take the limit of the above equation,

Sum = lim (500/800)ⁿ – 1

Sum = lim (500/800)⁸⁰⁰ – 1 = 0.35 – 1 = -0.65

Hence, the sum of harmonic progression from 500 to 800 is -0.65.

## Example

Let us consider an example to better understand the concept of the sum of harmonic progression.

Suppose we have an arithmetic progression of 7 terms, that is, 5,7,9,11,13,15 and 17.

The sum of this harmonic progression is given by the following formula:

Sum = ᵍⁿ ᶬᵐ – 1

Here, ᵍ = the first term in the series or 5 and n is the number of terms in the series or 7.

Thus, the sum of the harmonic progression of 5 to 17 is expressed as follows:

Sum = 5 ᶬᵐ7 – 1

Expanding this equation, we get:

Sum = (5/7)⁷ – 1

Now, if we take the limit of the above equation,

Sum = lim (5/7)ⁿ – 1

Sum = lim (5/7)⁷ – 1 = 0.21 – 1 = -0.79

Hence, the sum of harmonic progression from 5 to 17 is -0.79.

Table : Sum of Harmonic Progression from 5 to 17

|n | ᶬᵐ | Sum |
|——|——|——|
|5 | 1/7 | -0.79|

## Conclusion

The sum of harmonic progression is a formula used to calculate the sum of a harmonic series. In this article, we looked at the sum of harmonic progression from 500 to 800 terms. We also provided an example to understand the concept and discussed the formula used to calculate the sum.