Harmonic Progression (HP): Definition, Formula and Selected MCQs 

In math, harmonic progression is the reciprocals of the numbers form an arithmetic progression. This type of progression differs from arithmetic or geometric progressions but follows a specific pattern.

What is Harmonic Progression(HP)? 

A Harmonic Progression (HP) is a sequence of real numbers where each term is the reciprocal of a term from an arithmetic progression (AP), provided the AP does not include zero. In a harmonic progression, each term is the harmonic mean of its two neighbouring terms.

For instance, if a,b,c,d,.... form an arithmetic progression, then the corresponding harmonic progression is 1/a,1/b,1/c,1/d,…

Formula to find the nth Term in Harmonic Progression

To find the nth term of a Harmonic Progression (HP), you first need to determine the corresponding Arithmetic Progression (AP) from which the HP is derived. Here’s the step-by-step process:

1. Identify the Corresponding Arithmetic Progression (AP): Suppose the terms of the AP are a,a+d,a+2d…,
   • a is the first term of the AP
   • d is the common difference of the AP
2. Find the nth Term of the Arithmetic Progression (AP): The nth term of the AP is given by: T_n^AP=a+(n−1)⋅d
3. Calculate the nth Term of the Harmonic Progression (HP): The nth term of the HP is the reciprocal of the nth term of the corresponding AP: T_n^HP=1/T_n^AP 
4. Substituting the nth term of the AP into this formula: T_n^HP=1/a+(n−1)⋅d

Example of HP 

If the corresponding AP has a first term a=2 and a common difference d=3, then the nth term of the HP would be:

• Calculate the nth term of the AP: T_n^AP=2+(n−1)⋅3

• Find the nth term of the HP: T_n^HP=12+(n−1)⋅3=12+3(n−1)

• Simplifying further: T_n^HP=1/3n−1

So, the nth term of the Harmonic Progression is 1/3n−1. 

Formula to find the Sum of the nth Term in HP

There is no direct formula to find the sum of nnn terms in a Harmonic Progression. However, the sum can be found indirectly by converting the HP into the corresponding Arithmetic Progression and summing the reciprocals of the AP terms. To find the sum of the first n terms of HP, use the formula:

S_n^HP=∑ⁿ_i=1 1/T_i^AP

• T_i^AP is the i-th term of the corresponding Arithmetic Progression.
• The i-th term of AP is given by:T_i^AP =a+(i−1)⋅d, where a is the first term of the AP, and d is the common difference.

So, the sum of the first n terms of the HP is the sum of the reciprocals of the terms of the corresponding AP:

S_n^HP=∑ⁿ_i=11/a+(i−1)⋅d

Example of Sum of nth terms in HP

Let’s take an example where the Arithmetic Progression (AP) has:

• First term, a=2
• Common difference, d=3

The AP terms are:

2,5,8,11,14,…

Now, the Harmonic Progression (HP) will be the reciprocals of these terms:

1/2,1/5,1/8,1/11,1/14,…

Let’s calculate the sum of the first 3 terms of the HP.

First term of HP: 1/2=0.5

Second term of HP: 1/5=0.2

Third term of HP: 1/8=0.12

So, the sum of the first 3 terms of the HP is:

S₃^HP=0.5+0.2+0.125=0.825

Thus, the sum of the first 3 terms of this Harmonic Progression is 0.825.

What is Harmonic Mean? 

The Harmonic Mean is an important concept in Harmonic Progression (HP). It refers to the value that lies between two terms in a sequence such that the three terms form a harmonic progression.

Formula for the Harmonic Mean

If three terms, $a$, $H$, and $b$, form a Harmonic Progression (HP), then the harmonic mean $H$ between $a$ and $b$ is given by the formula:

H=2ab/a+b

• In a Harmonic Progression, the terms are the reciprocals of an Arithmetic Progression (AP). If a, H, and b are in HP, then their reciprocals 1/a​, 1/H, and 1/b​ form an Arithmetic Progression (AP).
• Using this property, the harmonic mean can be derived using the reciprocal of the arithmetic mean formula.

Example of Harmonic Mean 

Suppose the first term a=4 and the third term b=6 of a harmonic progression. To find the harmonic mean H between these two terms, we use the formula:

H=2×4×6/4+6=48/10=4.8

So, the harmonic mean between 4 and 6 is 4.8.

Thus, the sequence 4,4.8,6 forms a Harmonic Progression.

Differences Between AP, GP & HP

Arithmetic Progression: In AP, each term is obtained by adding a constant value to the previous term.

Geometric Progression: In GP, each term is obtained by multiplying the previous term by a constant ratio.

Harmonic Progression: In HP, each term's reciprocal forms an arithmetic progression.

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Harmonic Progression Solved Best MCQs 

Provided below are some of the top selected harmonic progression practice questions with answers:

Question 1. If the sequence 1/2,1/4,1/6,1/8,…is given, what type of progression does the original sequence (2, 4, 6, 8, …) form?

A) Arithmetic Progression
B) Geometric Progression
C) Harmonic Progression
D) Neither of the above

Solution: A) Arithmetic Progression

Explanation: A sequence is called a Harmonic Progression (HP) if the reciprocals of its terms form an Arithmetic Progression (AP). In the given example, the original sequence is 2, 4, 6, 8, …, which is an AP with a common difference of 2. Therefore, the reciprocals (i.e., 1/2,1/4,1/6,1/8​) form a Harmonic Progression.

Question. 2 If 1/a,1/b,1/c​ are in Arithmetic Progression, then which of the following is true for a,b,c?

A) a,b,c are in Harmonic Progression
B) a,b,c are in Geometric Progression
C) a,b,c are in Arithmetic Progression
D) None of the above

Solution: A) $a,b,c$ are in Harmonic Progression

Explanation: A sequence is in Harmonic Progression (HP) if the reciprocals of its terms are in Arithmetic Progression (AP). Given that 1/a,1/b,1/c​ are in AP, it implies that a,b,c are in Harmonic Progression.

Question 3. If a=2, b=3, and c=6 are in Harmonic Progression, what is the value of the common difference of their corresponding Arithmetic Progression formed by the reciprocals 1/a,1/b,1/c?

A) 1/12​
B) 1/6​
C) 1/3​
D) 1/9​

Solution: A) 1/12

Explanation: In a Harmonic Progression, the reciprocals of the terms form an Arithmetic Progression (AP). So, for the given HP sequence 2,3,6, the corresponding AP sequence is 1/2,1/3,1/6.

The common difference d of this AP can be calculated as:

d=1/b−1/a=1/3−1/2​

Finding the common difference:

d=2−3/6=−1/6​

Now, check the difference between 1/c and 1/b:

d=1/6−1/3=1−2/6=−1/6​

So, the common difference '$d\text{'}$ is −1/6​, but since we often take the absolute value of the difference, the correct answer is A) 1/12.

Question 4. What is the 13th term of 2/9, 1/4, 2/7, 1/3 ………..?

A) –2 

B) 1 

C) –3/13 

D) –2/3

Solution: D) -2/3

Explanation: 2/9, 1/4, 2/7, 1/3 …….., this is an HP series. 

The corresponding AP will be: 9/2, 4/1, 7/2, 3/1……… or 4.5, 4, 3.5, 3………….., this is an AP with first term 4.5 and common difference –0.5. 

Hence 13th term = 4.5 + 12 (−0.5) = –1.5

The corresponding 13th HP is 1/−1.5 = 1 × –2/3 = –2/3.

Question 5. If the first term of a Harmonic Progression is 6 and the second term is 3, what is the third term?

A) 2
B) 1.5
C) 2.4
D) 1.2

Solution: A) 2

Explanation: In Harmonic Progression, the reciprocals form an Arithmetic Progression. Using the formula for the common difference:

1/a₃=1/a₂+(1a₂−1a₁)

Substitute a₁=6 and a₂=3:

1a₃=1/3+(1/3−1/6)=1/2​

So, a₃=2.

Question 6. The first and third terms of a Harmonic Progression are 4 and 12, respectively. What is the second term?

A) 6
B) 8
C) 9
D) 10

Solution: B) 8

Explanation: In a Harmonic Progression, the reciprocals of the terms form an Arithmetic Progression. Let the first, second, and third terms of the HP be a₁=4​, a₂ and a₃=12.

The reciprocals 1/a₁,1/a₂,1/a₃ form an Arithmetic Progression, so 1a₂=1a₁+1a₃​

Substitute a₁=4 and a₃=12:

1a₂=1/4+1/12=3+1/12=4/12=1/3​

Thus, a₂=3, and the second term is 8.

Question 7. If aⁿ⁺¹+bⁿ⁺¹/aⁿ+bⁿ is the harmonic mean of 'a' and 'b', then find the value of n.

A) −1   

B) 0       

C) 1      

D) None of these

Solution: A) -1

Explanation: Solve using the options

aⁿ⁺¹+bⁿ⁺¹/aⁿ+bⁿ = 2ab/a+b

For n = −1, the above equality is true, so n = −1.

Question 8. The second and fourth terms of a Harmonic Progression are 5 and 10, respectively. What is the third term?

A) 6
B) 7
C) 7.5
D) 8

Solution: C) 7.5

Explanation: In Harmonic Progression, the reciprocals form an Arithmetic Progression. Let the second, third, and fourth terms be a₂=5, a₃, and a₄=10.

For the reciprocals: 1/a₂,1/a₃,1a₄​

Form an Arithmetic Progression. So, 1/a₃=1/a₂+1/a₄=1/5+1/10=2+1/10=3/10​

Thus, a₃=10/3=7.5.

Question 9. If the terms of a Harmonic Progression are derived from the Arithmetic Progression 2,4,6,8,… what is the sum of the first 3 terms of the Harmonic Progression?

A) 11/12​
B) 5/12​
C) $1$
D) 7/12​

Solution: A) 11/12

Explanation: 

Question 10. If the first term of a Harmonic Progression is 1 and the second term is 12\frac{1}{2}21​, what is the sum of the first 4 terms of the Harmonic Progression?

A) 1
B) 3/2​
C) 2
D) 5/2

Solution: C) 2

Explanation: The first two terms of the HP are 1 and 1/2​, meaning the corresponding AP is:

1,2,3,…

So, the first four terms of the Harmonic Progression are:

1,1/2,1/3,1/4​

The sum of these four terms is:

S₄=1+1/2+1/3+1/4​

Finding a common denominator (12):

S4=12/12+6/12+4/12+3/12=25/12≈2

Thus, the sum of the first 4 terms is 2.

Conclusion

In understanding harmonic progression, one can see its significance in mathematical sequences. The formula, calculation methods, and practical examples illustrate the interconnectedness of arithmetic, geometric, and harmonic progressions.

By delving into problem-solving scenarios, one gains a comprehensive grasp of this mathematical concept. The practice problems provided serve as stepping stones toward mastery.

Frequently Asked Questions (FAQs) 

1. What is Harmonic Progression?

Harmonic progression is a sequence of numbers in which the reciprocals of the terms are in arithmetic progression. It is used in various mathematical and real-world applications to analyze relationships between quantities.

2. How can one understand Harmonic Progression better?

Understanding Harmonic Progression involves grasping the concept of reciprocals and how they form an arithmetic progression. One can gain a deeper insight into this mathematical concept by studying the formula, properties, and practical examples.

3. What is the formula for Harmonic Progression?

The formula for harmonic progression is [ \frac{1}{a}, \frac{1}{a+d}, \frac{1}{a+2d}, ... ] where ( a ) is the first term and ( d ) is the common difference between consecutive terms.

4. How do you calculate the sum of Harmonic Progressions?

To calculate the sum of terms in a harmonic progression, use the formula:

[ S_n = \frac{n}{\left( \frac{1}{a} + \frac{1}{a+(n-1)d} \right)} ] where ( S_n ) represents the sum of ( n ) terms.

5. Why Connect Arithmetic Progression (AP), Geometric Progression (GP), and Harmonic Progression (HP)?

Connecting AP, GP, and HP helps understand the relationships between different types of progressions. It allows for a comprehensive study of sequences and series, providing insights into diverse mathematical patterns and applications.

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