In Mathematics, a sequence is a list of numbers (ascending, descending, or ascending and descending based on the pattern) that follows a predictable, precise pattern. Some easy sequences include … Show
1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21 … (odd numbers) 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22 … (wow, even numbers) 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 … (the Fibonacci sequence) More complex math sequences would be like … 23, 21, 24, 19, 26, 15, 28, 11, 30, 7, 36 (alternate subtracting then adding consecutive prime numbers starting with 2) 2, 9, 3, 18, 4, 36, 5, 72, 6, 144, 7 (+1 to odd terms, x2 to even terms) 3, 8, 15, 24, 35, 48, 63, 80, 99, 120, 143 (starting with 1x3, each term increases the two factors by 1) Notice that you need at least three terms to confirm the sequence/pattern and having four or more available make the sequence much more obvious. This is also true for musical sequences both melodic and harmonic. Before we understand Harmonic Sequence or Harmonic Series, we must understand what is Arithmetic sequence /Arithmetic Progression. I assume you all have already covered Arithmetic Progression under Sequence and Series. Here we will understand every concept of Harmonic Series following the Arithmetic sequence. Also read: What is a harmonic series? The reciprocal form of the Arithmetic Sequence with numbers that can never be 0 is called Harmonic Sequence. And the sum of such a sequence is known as Harmonic Series If we have Arithmetic Sequence as 4,6,8,10,12 with the common difference of 2 i.e. d =2 The Harmonic Sequence of the above Arithmetic Sequence is 1/4, 1/6, 1/8,1/10,1/12…. Let’s take another example We have to determine if the below series is Harmonic series or not 3/7,1/3,3/11,3/13,3/15…. Now if we prove that the reciprocal of the above sequence is A.P with a common difference then we can establish that the sequence is the Harmonic sequence . And the sum of this sequence would be a harmonic series. We must first understand what arithmetic sequence or arithmetic progression is before learning about harmonic sequence or harmonic series. An arithmetic series is a set of numbers where the difference between any two consecutive elements is always constant. Arithmetic Progression, Geometric Progression, and Harmonic Progression are three forms of progression. In this article, we are going to discuss the harmonic sequence ,harmonic progression in maths and its formula along with solved examples. Harmonic Sequence DefinitionThe harmonic sequence in mathematics can be defined as the reciprocal of the arithmetic sequence with numbers other than 0. The sum of harmonic sequences is known as harmonic series. It is an infinite series that never converges to a limit. For example, let’s take an arithmetic sequence as 5, 10, 15, 20, 25,... with the common difference of 5. Then its harmonic sequence is: 1/5, 1/10, 1/15,1/20,1/25…. Harmonic Progression in MathsIn Mathematics, we can define progression as a series of numbers arranged in a predictable pattern. It's a form of number set that adheres to strict, predetermined laws, which is the main difference between a progression and a sequence, as a sequence is solely based on specific logical rules. A Harmonic Progression (H.P.) is a series of real numbers that can be determined by multiplying the reciprocals of an arithmetic progression that doesn't contain zero. For example, for the arithmetic progression, p, q, r, s,... The harmonic progression is: 1/p, 1/q, 1/r, 1/s,... What does Harmonic Mean?The inverse of the arithmetic mean of the reciprocals is used to measure the harmonic mean. So, if 1/a and 1/b are two consecutive terms, their harmonic mean is given by : \[ H = \frac{2ab}{a+b} \] The following is the formula for calculating the harmonic mean for n terms: Harmonic Mean = \[ \frac{n}{ [(\frac{1}{a})+(\frac{1}{b})+(\frac{1}{c})+(\frac{1}{d})+...] } \] Where a, b, c, d are the values and n is the total number of values present. First-Term of Harmonic ProgressionThe first term of the harmonic progression is denoted by a. The sum of the series can never be an integer except for the first term, as it can be 1. Common Difference of Harmonic ProgressionThe common difference is the difference between any two consecutive numbers in the series. The common difference is denoted as ‘d’ and it is the same in any progression. Example of First Term and Common Difference of H.P.If 1/a, 1/b, 1/c are three terms in Harmonic Progression, then the first term is 1/a and the common difference is d. So, the common difference will be, \[ d = \frac{1}{a} - \frac{1}{b} = \frac{1}{c} - \frac{1}{b} \] or \[ \frac{a-b}{ab} = \frac{b-c}{bc} \] or \[ \frac{a}{c} = \frac{b-c}{a-b} \] Harmonic Progression FormulaIn an H.P with n terms, we need the formula to find the value of its nth term. This formula is equal to the reciprocal of the formula for finding the nth term of arithmetic progression. Thus, the formula to find the nth term of the harmonic progression series is given below: nth term of the Harmonic Progression = \[ \frac{1}{[a + (n−1)]} \times d \] Where “a” is the first term of H.P. “d” is the common difference “n” is the number of terms in H.P. Therefore, we can say that the nth term of H.P = \[\frac{1}{(n^{th} \text{ term of the corresponding A.P)}}\] Sum of Harmonic ProgressionThe sum of n terms in a harmonic progression can be determined easily if the first term and the value of n terms is known. If the terms \[ \frac{1}{a}, \frac{1}{a+d} , \frac{1}{a+2d}..., \frac{1}{a+(n-1)d} \] make a harmonic progression, the formula to find the sum of n terms in the harmonic progression can be obtained by the formula: Sum of n terms, \[ S_{n} = \frac{1}{d}\] In \[ \frac{2a+(2n-1)d}{2a-d} \] Where, “a” is the first term of H.P. “d” is the common difference of H.P. “ln” is the natural logarithm Properties of Harmonic Progression
\[ \frac{1}{H-a} + \frac{1}{H-b} = \frac{1}{a} + \frac{1}{b} \] (H - 2a)(H - 2b) = H2 \[ \frac{H+a}{H-a} + \frac{H+b}{H-b} = 2 \]
Uses of Harmonic Sequence in Real LifeSome important applications of harmonic progression in everyday life include:
Did You Know?The harmonic sequence was first studied back in the 6th century by the Greek mathematician Pythagoras. He first used harmonic progression to study the nature of the universe and also to study music. Also, a trick to solve harmonic progression questions easily is to convert H.P into arithmetic progression whenever possible. What is the harmonic sequence formula?Harmonic Mean: In a harmonic progression, any term of the series is the harmonic mean of its neighboring terms. Harmonic Mean = n /[1/a + 1/(a + d)+ 1/(a + 2d) +1/(a + 3d) +….] Harmonic mean of two terms a and b = (2ab) / (a + b).
What is harmonic sequence and examples?Harmonic Sequence Definition
It is an infinite series that never converges to a limit. For example, let's take an arithmetic sequence as 5, 10, 15, 20, 25,... with the common difference of 5. Then its harmonic sequence is: 1/5, 1/10, 1/15,1/20,1/25….
What is the 10th term in the harmonic sequence?The tenth term would be 4/11. A harmonic progression is formed by taking the reciprocals of an arithmetic progression. The reciprocals of the four elements listed above would be 20/4, 19/4, 18/4, and 17/4, which form an arithmetic progression in which element is equal to the previous element - 1/4.
Why do you think is called harmonic sequence?Why is the series called "harmonic"? form an arithmetic progression, and so it is that a sequence of numbers whose inverses are in arithmetic progression is said to be in harmonic progression. Pythagoras mixed his mathematics and physics with a liberal helping of mystical mumbo-jumbo.
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