Content Curator
Harmonic progression is a mathematical progression in which there is a set of rules to find the nth number in the progression and it is determined by taking the reciprocals of the arithmetic progression in which none of the numbers is 0.
- In harmonic progression, the reciprocals of the series would be in arithmetic progression.
- A progression in mathematics is a sequence of numbers organized in an ordered manner.
- It is a type of number set that follows definite and specific rules.
- There is a difference between progression and sequence.
- A progression uses a particular formula to calculate its nth term, whereas a sequence is based on certain logical rules.
- A progression is commonly classified into three types: arithmetic progression, geometric progression, and harmonic progression.
| Table of Content |
Key Terms: Harmonic progression, Arithmetic progression, Geometric progression, Reciprocals, Harmonic mean, Integers, Progression and sequence
What is Harmonic Progression?
[Click Here for Sample Questions]
A sequence of real numbers that is obtained by taking the reciprocals of the arithmetic progression that does not contain zero is known as a harmonic progression (HP).
- Any term in the sequence is regarded as the harmonic mean of its two neighbors in a harmonic progression.
- Let the sequence a, b, c, d,….. be considered as the arithmetic progression.
- Then the harmonic progression will be given by 1/a, 1/b, 1/c, 1/d,…….
Harmonic Mean
The reciprocal of the arithmetic mean of the reciprocals is used to calculate the harmonic mean.
The formula of harmonic mean is given by
\(Harmonic\:\:Mean = \frac {n}{\frac {1}{a}+\frac {1}{b}+ \frac {1}{c}+....}\)
Where
- a, b, c,…. are the numbers
- n is the total count of the numbers
Also Read:
| Related Articles | ||
|---|---|---|
| Straight Lines | Angle Between two lines | Horizontal and Vertical Lines |
| Real-Valued Function | Trigonometric Functions | Complex numbers and Quadratic Equations |
Harmonic Progression Formula
[Click Here for Sample Questions]
We must determine the corresponding arithmetic progression sum in order to solve the harmonic progression problems. This indicates that the reciprocal of the nth term in the arithmetic progression is equal to the nth term of the harmonic progression.
The formula to determine the nth term of the harmonic progression series is given by
\(The \:n^{th} \:term\: of \:the\: Harmonic\: Progression\: (H.P)=\frac {1}{a+(n-1)d}\)
Where,
- ‘a’ is the first term
- ‘n’ is the number of terms
- ‘d’ is the common difference between the terms
The above formula can also be written as
\(The \:n^{th} \:term\: of \:the\: Harmonic\: Progression\: (H.P)=\frac {1}{n^{th}\: term \:of \:the\: corresponding \:A.P}\)
Harmonic Progression Sum
[Click Here for Sample Questions]
Let a harmonic progression sequence is given by
1/a, 1/a+d, 1/a+2d, …., 1/a+(n-1)d
Then the formula of the sum of n terms in the harmonic progression is given by
\(S_n = \frac {1}{d}\: \ln {\frac {2a+(2n-1)d}{2a-d}}\)
Where
- a is the first term of A.P
- d is the common difference of A.P
-
ln is the natural logarithm
Arithmetic Progression
[Click Here for Sample Questions]
A series of numbers in which the second number is obtained by adding a fixed number to the first one for each pair of consecutive terms is known as an arithmetic sequence or progression.
The formula of arithmetic progression to find nth term is given by
an = a + (n − 1) × d
Where
- an is the nth term
- a is the first term
- d is the common difference
- n is the number of terms
Geometric Progression
[Click Here for Sample Questions]
A geometric progression, also referred to as a geometric sequence, is a series of numbers where each number differs from the other by a common ratio.
The formula to find the nth term of GP is given by
an = arn-1
Where
- an is the nth of the GP
- a is the first term
- r is the common ratio
- n is the total number of terms
Relation Between AP, GP, and HP
The relation between the Arithmetic, Geometric, and Harmonic Means, denoted by the letters A.M., G.M., and H.M., for any two numbers, is as follows:
G.M2 = A.M × H.M, where A.M, G.M, H.M are in G.P
A.M ≥ G.M ≥ H.M
Things to Remember
- In harmonic progression, the reciprocals of the series would be in arithmetic progression.
- A progression in mathematics is a sequence of numbers organized in an ordered manner.
- There is a difference between progression and sequence.
- A harmonic progression (HP) is a series of real numbers that is produced by taking the reciprocals of the arithmetic progression that does not contain zero.
- The reciprocal of the arithmetic mean of the reciprocals is used to calculate the harmonic mean.
- A geometric progression, also referred to as a geometric sequence, is a series of numbers where each number differs from the other by a common ratio.
Also Read:
Sample Questions
Ques. What is harmonic progression? (2 Marks)
Ans. A harmonic progression can be formed by taking the reciprocals of an arithmetic progression. A sequence is equivalent to a harmonic progression if each term is the harmonic mean of its neighbors.
Ques. Define the 6th and 9th term of the H.P 6, 4, 3,… (3 Marks)
Ans. H.P = 6,4,3
A.P = 1/6, 1/4, 1/3
d = 1/4 -1/6 = 1/12
6th term= 1/ [a + (n – 1)d]
= 1/ [1/4 + (6-1)1/12]
= 1/ [7/12]
=12/7
9th term= 1/ [a + (n – 1)d]
=12/10
Ques. Find the 4th and 5th terms of the A.P 4, 8, 12,… (3 Marks)
Ans. For A.P, nth term = an = a + (n – 1) d
a = 4
d = 8 – 4 = 4
a4 = 4 + (4 – 1)4
= 16
Similarly, a5 = 4 + (5 – 1)4
=4 + 16
=20
Ques. Find the arithmetic progression up to 5 terms if the first term is 3 and the common difference is 4. (3 Mark)
Ans. Since, an+1 = an+d
Given
- a= 3
- d=4
Therefore, a2 = 3 + 4 = 7
a3 =7 + 4 = 11
a4 =11 + 4 = 15
a5 = 15 + 4 = 19
Therefore, the A.P is 3,7,11,15,19
Ques. In an A.P. if mth term is n and the nth term is m, where m ≠ n, find the pth term. (3 Marks)
Ans. We have am = a + (m – 1) d = n, ... (1)
and an = a + (n – 1) d = m ... (2)
Solving (1) and (2), we get (m – n) d = n – m, or d = – 1, ... (3)
and a = n + m – 1 ... (4)
Therefore ap= a + (p – 1)d = n + m – 1 + ( p – 1) (–1) = n + m – p
Hence, the pth term is n + m – p
Ques. Find the whole arithmetic progression if there are 4 terms between 4 and 29. (2 Marks)
Ans. Given,
- a = 4
- a6 = 29
Therefore, a6 = a + (n – 1)d
⇒ 29 = 4 + 5d
Therefore, d = 5
Now, through an = a + (n – 1)d
A.P. is 4,9,14,19,24 and 29.
Ques. Find the 4th term of the G.P 6, 36, 216.. (2 Marks)
Ans. For G.P, an = arn-1
Here,
- a = 6
- r= 36/6= 6
Therefore, a4 = 6(6)4-1
= 1296
Ques. Find the first 4 terms between the G.P. if the first term is 25 and 3rd term is 625. (2 Marks)
Ans. Given, a = 25
a3 = 625
⇒ a3 = 25(r)3-1
⇒ 625 = 25(r)2
Therefore, r = 5
The required GP = 25, 125, 625, 3125
Ques. In a G.P. the 3rd term is 24 and the 6th term is 192. Find the 10th term. (2 Marks)
Ans. Here, a3 = ar2 = 24 ... (1)
and a6 = ar5 = 192 ... (2)
Dividing (2) by (1), we get r = 2.
Substituting r = 2 in (1), we get a = 6.
Hence a10 = 6 (2)9 = 3072
Ques. Define the harmonic progression up to 3 terms if the first term is 2 and the common difference in an A.P is 4. (2 Marks)
Ans. Since, a = 2 and d = 4
Therefore, A.P is an = a + (n – 1) d
Therefore, a2 = 2 + 4 = 6
a3 = 2 + 8 = 10
A.P = 2, 6,10
Thus, H.P = 1/2, 1/6 , 1/10
Ques. Find the A.P if the H.P is 1/5, 1/10, 1/15 .. (1 Mark)
Ans. A.P is the reciprocal of the H.P
Therefore, A.P is 5,10,15…
For Latest Updates on Upcoming Board Exams, Click Here: https://t.me/class_10_12_board_updates
Check-Out:
Comments