Sum of Harmonic Progression Formula is reciprocal of the sum of AP. In this article, we will learn what is a harmonic series, Sum of Harmonic Progression with Formula, Derivation, the sum of infinite harmonic progression formula and Solved Examples

A sequence of numbers is said to be a hp in maths if the reciprocal of the terms are in AP. In simple terms, we can say that if a,b,c,d,e,f is in AP then the harmonic progression can be written as 1/a, 1/b, 1/c, 1/d, 1/e, 1/f.

Then the harmonic sequence is as follows:

\(\frac{1}{a},\ \frac{1}{a+d},\ \frac{1}{a+2d},\ \frac{1}{a+3d},\dots\)

• First term = a
• Common difference = d

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How to Find Sum of Harmonic Progression

Harmonic Progressions are reciprocals of Arithmetic Progression. Hence, the sum of harmonic progressions is also the reciprocal of the sum of the arithmetic progression. Sum of arithmetic progression is given by \(S_n = {n\over2}[2 a_1 + (n – 1)d]\)

Sum of ‘n’ terms of given HP is \(\frac{2}{n} \left [ \frac{1}{2a + (n-1))d} \right ]\)

Sum of n terms in HP;

\(\text{For }\frac{1}{a},\frac{1}{a+d},\frac{1}{a+2d},\dots.,\frac{1}{a+(n-1)d}\)

\(S_n=\frac{1}{d}\ln\left(\frac{2a+\left(2n−1\right)d}{2a−d}\right)\)

Where:

• “a” is the first term of A.P
• “d” is the common difference of A.P
• “ln” is the natural logarithm

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Sum of Infinite Harmonic Progression

In mathematics, the harmonic series is the divergent infinite series

\({\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n}}=1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+{\frac {1}{5}}+\cdots .}\)

The sum of infinite harmonic progression is as follows:

by 

\(\sum_{k = 1}^{\infty}{1\over{k}}=1+{1\over{2}}+{1\over{3}}+{1\over{4}}+…\)

Infinite harmonic progressions are not summable. This series does not converge but rather diverges.

Relation Between AP, GP, and HP 

In maths, we often come across three types of number patterns or sequences:
Arithmetic Progression (AP), Geometric Progression (GP), and Harmonic Progression (HP). These are just different ways numbers can increase or decrease in a certain pattern.

When we take any two numbers, we can find three types of averages (means) from them:

1. Arithmetic Mean (A.M):

This is the simple average of two numbers.
Formula: A.M = (a + b) / 2
It adds both numbers and divides by 2. It's like the middle value between the two.

2. Geometric Mean (G.M):

This is found by multiplying the numbers and taking the square root.
Formula: G.M = √(ab)
It is useful when the numbers are in multiplication or ratio patterns.

3. Harmonic Mean (H.M):

This mean is best used when dealing with speed, rates, or ratios.
Formula: H.M = (2ab) / (a + b)
It gives more weight to smaller numbers.

Important Relationship:

The three means are connected by a special formula:
(G.M)² = A.M × H.M
This means if you square the geometric mean, you'll get the same result as multiplying the arithmetic and harmonic means.

Also, for any two positive numbers, the order of the means is always:
A.M ≥ G.M ≥ H.M
So, the arithmetic mean is the largest, the geometric mean is in the middle, and the harmonic mean is the smallest.

These means also form a Geometric Progression (G.P), which means each one is related to the next by the same ratio.

Sum of Harmonic Progression Examples

1.If the sum of reciprocals of the first 11 terms of an HP series is 110, find the 6th term.

A.

Reciprocals of the first 11 terms will be AP

Therefore, Sn = n /2 [2a + (n − 1) d]

S(11)=110 (putting n =11)

110 = 11/2[2a + (11 − 1) d]

110 = 11/2 x [2a + 10d]

220 = 22a + 110d

22a + 110d = 220

a + 5d = 10, which is the 6th term of the AP series

Therefore, the 6th term in HP = 1/10

2.Find the sum of the harmonic progression 1/6, 1/12, 1/18, 1/24.

A.

Given that, 1/6, 1/12, 1/18, 1/24 are in H.P.

To find the sum of an H.P., we directly add the terms, because the sum of an H.P. is not the reciprocal of the sum of the corresponding A.P.

Convert each term to have a common denominator:

• LCM of 6, 12, 18, and 24 is 72

Now express each fraction with denominator 72:

• 1/6 = 12/72
• 1/12 = 6/72
• 1/18 = 4/72
• 1/24 = 3/72

Now add them:

Sum = (12 + 6 + 4 + 3) / 72 = 25 / 72

The sum of the H.P. is: 25/72

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