A Harmonic Progression, also known as a Harmonic Sequence, is a series of real numbers generated by taking the reciprocals of an Arithmetic Progression or Arithmetic Sequence.
A Harmonic Progression is a sequence of values where the reciprocals of these values form an Arithmetic Progression. There are specific formulas for finite Harmonic Progressions. In this sequence, each term is the Harmonic Mean of its adjacent terms, making it a series of real numbers derived by taking the reciprocals of an Arithmetic Progression and calculating the mean of its neighboring terms.
HP is Represented in the form of
In this Page Harmonic Progression Formulas is given that is useful to Solve many Problems in different Competitive Examinations.
Example 1: In a harmonic progression, the sum of the first 5 terms is 6, and the sum of their cubes is 405. What is the first term of this progression?
(a) (1/3)
(b) (1/4)
(c) (1/5)
(d) (1/6)
Ans: (a)
Let a be the first term and d be the common difference of the harmonic progression. The sum of the first 5 terms is given by ((S_{5}) = 5a + 10d = 6). The sum of their cubes is given by (S_{cubes}) = a^{3} + (a + d)^{ 3} + (a + 2d)^{3} + (a + 3d)^{3} + (a + 4d)^{3} = 405). Using the value of ((S_{5})), we can solve for d, and then using ((S_{cubes} )), we can solve for a, which turns out to be (1/3).
Example 2: In a harmonic progression, the sum of the first 6 terms is 3 times the sum of their reciprocals. What is the sum of the first 12 terms of this progression?
(a) 2
(b) 3
(c) 4
(d) 5
Ans: (c)
Let (S_{n}) be the sum of the first n terms and (S_{n}′) be the sum of the reciprocals of the first n terms. The given condition can be written as ((S_{6}) = (3S_{6}′)). Using the formula for the sum of the first n terms of a harmonic progression, we have
Example 3: The sum of an infinite harmonic progression is (5/3). What is the sum of the squares of the terms in this progression?
(a) (10/3)
(b) (25/9)
(c) (15/4)
(d) (9/5)
Ans: (a)
The sum of the squares of the terms in a harmonic progression is given by where a_{1} is the first term. Given the sum of the infinite harmonic progression as (5/3), we can use the result
Example 4: In a harmonic progression, . What is the sum of the first 10 terms of this progression?
(a) (185/99)
(b) (99/185)
(c) (135/70)
(d) (70/135)
Ans: (a)
The sum of the first n terms of a harmonic progression is given by S_{n} = a_{1}
where a_{1} is the first term. Substituting the given ℎ_{nth }term, we get
Example 5: The sum of the first n terms of a harmonic progression is given by What is the common difference between the terms of this progression?
(a) (1/3)
(b) (2/3)
(c) (3/4)
(d) (4/5)
Ans: (a)
The sum of the first n terms of a harmonic progression is given by where a is the common difference. Comparing this with the given formula, we have
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