Harmonic Sequence Calculator

A sequence of numbers that are reciprocals of an arithmetic sequence that does not contain 0 is called the harmonic sequence. Also, it is known as harmonic progression denoted as H.P in short. In HP, any term in the sequence is acknowledged as the harmonic means of its two neighbours. 

For instance, if the sequence of the first term is a and it is in arithmetic progression then the formula to find the harmonic sequence is H.P = 1/a, 1/a + d, 1/a+2d, 1/a+(n-1)d, … .,1/a+(n-1)d,… .

What is Harmonic Mean? 

The reciprocal of the arithmetic mean is called harmonic mean. To calculate the harmonic mean, we will use the formula of the harmonic mean which 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.

First-term of Harmonic Progression

For any numbers series, the first term of the Harmonic progression is very crucial and it is denoted as 'a'. However, the sum of the series can never be an integer except for the first term as 1.

Common Difference of Harmonic Sequence

The difference between any two-consecutive number in the series is the same then it is known as the common difference. It is represented as 'd'.

Nth Term of H.P or Harmonic Sequence Formula

For calculating the problems of harmonic progression, we have to find the sum of corresponding arithmetic progression. In other words that the nth term of the harmonic progression is equal to the reciprocal of the nth term of the corresponding A.P. Thereby, the formula to obtain the nth term of the harmonic series and the harmonic sequence is given as:

nth_term = 1/(First term+(Total terms-1)*Common difference)

a_n = 1/(a+(T_Total-1)*d)

where,

• Nth term and is denoted by 'a_n' symbol.
• First Term - First-term is the initial term of a series or sequence. It is represented as 'a'. 
• Total Terms - the total number of terms in a particular series.
• Common difference - the difference between two successive terms of a harmonic progression. It is denoted by 'd'.

How to Find the Sum of Harmonic Sequence? 

In order to find the summation of harmonic sequence, you just need to apply the sum of harmonic sequence formula which is given below. Also, for harmonic progression, harmonic sequences summation can be solved easily in case you are aware of the first term and the total terms.

Sum of first n terms = 1/a + 1/(a + d) + 1/(a + 2d) + … +1/ [a + (n – 1) × d]

Remember that, we can also say n refers to infinity ∞

Next, the generic formulae for the nth term of Harmonic sequence is the reciprocal of A.P.

Thus, the sum of ‘n’ terms of HP is the reciprocal of A.P ie.,

Sum of AP is S_n = n/2[2a + (n − 1) × d]

Hence, The formula for Sum of HP is S_n = 2/n[2a + (n − 1) × d]

Apply the above sum of harmonic sequence formula and calculate the needed output manually with ease. 

Refer to SequenceCalculators.com and grab the opportunity of calculating harmonic sequence problems and other sequence and series-related complex questions at a faster pace. 

Example: 

Find the sum of the Harmonic Sequence: 1/12 + 1/24 + 1/36 + 1/48 + 1/60.

Solution:

Given sequence is 1/12 + 1/24 + 1/36 + 1/48 + 1/60

Firstly, solve the given series in AP

In AP, 12, 24, 36, 48, 60 is the sequence. 

From the above arithmetic progression, the first term (a) is 12, d is 12, n is 5

The sum of the harmonic sequence formula is the reciprocal of the sum of an arithmetic sequence. 

Thus, the formula of AP summation is S_n  = n/2[2a + (n − 1) × d]

Substitute the known values in the above formula

S_n = 5/2[2x12 + (5-1) X 12] = 180. 

Hence, the sum of 5 terms of H.P is reciprocal of A.P is 1/180. 

Example for Finding nth term of Harmonic Sequence

 

 

Example for Finding Sum of n terms of Harmonic Sequence

 

 

• To solve a problem on Harmonic Progression, one should make a similar AP series and then solve the problem.
• As the nth term of an A.P is given by a_n = a + (n-1)d, So the nth term of an H.P is given by 1/ [a + (n -1) d].
• For two numbers, if A, G, and H are respectively the arithmetic, geometric and harmonic means, then
  A ≥ G ≥ H
  A H = G2, i.e., A, G, H are in GP
• If we want to find three numbers in an H.P. then they need to be assumed as 1/a–d, 1/a, 1/a+d
• The majority of the questions of H.P. are solved by first converting them into A.P

1. What is Harmonic sequence formula?

A sequence of reciprocals of the arithmetic progression that does not contain 0 is called Harmonic Sequence or Harmonic Progression (HP). The formula to find the harmonic sequence is the formula of the harmonic mean (HM) =  n /[(1/a) + (1/b)+ (1/c)+(1/d)+….].

2. How do you find the nth term of a harmonic sequence?

To find the nth term of Harmonic sequence one should know the process of solving the nth term of the AP series. As HP of the nth term is the reciprocal of the nth term of Arithmetic progression. So, the formula of the nth term of Harmonic series is given by 1/ [a + (n -1) d]. Make use of this formula and solve the nth term of a harmonic sequence. Harmonic sequences formula can give absolute results.

3. How do you calculate harmonic series?

The sum from n = 1 to infinity with the terms 1/n is known as the harmonic series. In case you have addressed the first few terms then the series unfolds as 1 + 1/2 + 1/3 + 1/4 + 1/5 +. . .etc. Where n tends to infinity, 1/n tends to 0.

4. How do you find the common difference in harmonic sequences?

You can easily find the common difference in harmonic sequences with the help of sequencecalculators.com's Harmonic Sequence Calculator online for free of charge.