Q. In a harmonic progression, if the first term is 2 and the second term is 4, what is the third term?
A.
1
B.
3
C.
6
D.
8
Solution
In a harmonic progression, the reciprocals of the terms form an arithmetic progression. The reciprocals of 2 and 4 are 1/2 and 1/4. The common difference is -1/4. Therefore, the third term's reciprocal is 1/4 - 1/4 = 0, which means the third term is 1.
Q. In a harmonic progression, if the first term is 2 and the second term is 4/3, what is the third term?
A.
1
B.
3/2
C.
2/3
D.
1/2
Solution
In a harmonic progression, the reciprocals of the terms form an arithmetic progression. The reciprocals of the first two terms are 1/2 and 3/4. The common difference is 1/4, so the reciprocal of the third term is 1/2 + 1/4 = 3/4. Therefore, the third term is 1/(3/4) = 4/3.
Q. In a harmonic progression, if the first term is 3 and the second term is 6, what is the common difference of the corresponding arithmetic progression?
A.
1
B.
2
C.
3
D.
4
Solution
The reciprocals of the first two terms are 1/3 and 1/6. The common difference is 1/6 - 1/3 = -1/6, which is incorrect. The correct common difference is 1/3 - 1/6 = 1/6.
Q. In a harmonic progression, if the first term is 3 and the second term is 6, what is the third term?
A.
9
B.
12
C.
15
D.
18
Solution
The reciprocals of the terms are 1/3 and 1/6. The common difference is (1/6 - 1/3) = -1/6. The third term's reciprocal will be 1/6 - 1/6 = 0, which means the third term is 1/12, thus the answer is 12.
Q. In a harmonic progression, if the first term is 4 and the second term is 2, what is the common difference of the corresponding arithmetic progression?
A.
1
B.
2
C.
3
D.
4
Solution
The reciprocals of the terms are 1/4 and 1/2. The common difference is 1/2 - 1/4 = 1/4.
Q. In a harmonic progression, if the first term is 4 and the second term is 8, what is the third term?
A.
12
B.
16
C.
20
D.
24
Solution
The reciprocals are 1/4 and 1/8. The common difference is 1/8 - 1/4 = -1/8. The third term's reciprocal will be 1/8 - 1/8 = 0, hence the third term is 16.
Q. In a harmonic progression, if the first term is 4 and the second term is 8, what is the common difference of the corresponding arithmetic progression?
A.
1
B.
2
C.
3
D.
4
Solution
The reciprocals are 1/4 and 1/8. The common difference is 1/8 - 1/4 = -1/8, which means the common difference of the corresponding arithmetic progression is 1/8.
Q. In a harmonic progression, if the first term is 5 and the common difference of the corresponding arithmetic progression is 2, what is the second term?
A.
2.5
B.
3.33
C.
4
D.
6
Solution
The first term is 5, and the second term in the harmonic progression corresponds to the reciprocal of the second term in the arithmetic progression, which is 5 + 2 = 7. Thus, the second term is 1/7.
Q. In a harmonic progression, if the first term is 5 and the second term is 10, what is the common difference of the corresponding arithmetic progression?
A.
1
B.
2
C.
3
D.
5
Solution
The reciprocals are 1/5 and 1/10. The common difference is 1/10 - 1/5 = -1/10, which is the difference in the arithmetic progression.
Q. In a harmonic progression, if the first term is a and the second term is b, what is the formula for the nth term?
A.
1/(1/n + 1/a)
B.
1/(1/n + 1/b)
C.
1/(1/a + 1/b)
D.
1/(1/a - 1/b)
Solution
The nth term of a harmonic progression can be expressed as 1/(1/a + (n-1)d) where d is the common difference of the corresponding arithmetic progression.
Understanding Harmonic Progression (HP) is crucial for students preparing for various school and competitive exams. This mathematical concept not only appears frequently in exam papers but also forms the basis for many advanced topics. Practicing MCQs and objective questions on Harmonic Progression helps reinforce your understanding and boosts your confidence, ensuring you score better in your exams.
What You Will Practise Here
Definition and properties of Harmonic Progression (HP)
Formulas related to Harmonic Progression
Relationship between HP and Arithmetic Progression (AP)
Sum of the first n terms in a Harmonic Progression
Common applications of Harmonic Progression in real-life scenarios
Conversion between different types of progressions
Sample problems and practice questions on Harmonic Progression
Exam Relevance
Harmonic Progression is an important topic in various examinations, including CBSE, State Boards, NEET, and JEE. Students can expect questions that test their understanding of the definitions, properties, and applications of HP. Common question patterns include multiple-choice questions that require students to identify the correct formula or to solve problems involving the sum of terms in a Harmonic Progression.
Common Mistakes Students Make
Confusing Harmonic Progression with Arithmetic and Geometric Progressions.
Incorrectly applying the formulas for the sum of terms.
Overlooking the relationship between different types of progressions.
Failing to simplify expressions before solving problems.
FAQs
Question: What is a Harmonic Progression? Answer: A Harmonic Progression is a sequence of numbers where the reciprocals of the terms form an Arithmetic Progression.
Question: How do I find the nth term of a Harmonic Progression? Answer: The nth term of a Harmonic Progression can be found using the formula: \( \frac{1}{a_n} = \frac{1}{a} + (n-1)d \), where \( a \) is the first term and \( d \) is the common difference of the corresponding AP.
Now is the time to enhance your understanding of Harmonic Progression. Dive into our practice MCQs and test your knowledge to ensure you are well-prepared for your exams!
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