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What is H. M. of 1, 1/2, 1/3, . 1/n? (1) n (2) 2n (3) 2/(n 1) (4) n(n 1)/2?
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What is H. M. of 1, 1/2, 1/3, . 1/n? (1) n (2) 2n (3) 2/(n 1) (4) n(n ...
Solution:


Harmonic Mean (H.M.)

The Harmonic Mean (H.M.) of a set of numbers is the reciprocal of the arithmetic mean of the reciprocals of the numbers.


Let's find the H.M. of 1, 1/2, 1/3, ..., 1/n.


Solution Steps


  • Step 1: Find the reciprocal of each number in the set.


    • Reciprocal of 1 = 1

    • Reciprocal of 1/2 = 2

    • Reciprocal of 1/3 = 3

    • ...

    • Reciprocal of 1/n = n


  • Step 2: Find the arithmetic mean of the reciprocals.

  • Arithmetic Mean = (1 + 2 + 3 + ... + n) / n

    Arithmetic Mean = n(n + 1) / 2n

  • Step 3: Find the reciprocal of the arithmetic mean.

  • H.M. = reciprocal of Arithmetic Mean

    H.M. = 2n / (n + 1)



Answer:

The Harmonic Mean (H.M.) of 1, 1/2, 1/3, ..., 1/n is 2n / (n + 1).


Option (3) 2/(n+1)

Therefore, the correct answer is option (3) 2/(n+1).


Explanation:

In Step 3, we found that the H.M. of the set is 2n / (n + 1). Option (3) 2/(n+1) matches our answer, hence is the correct option.
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What is H. M. of 1, 1/2, 1/3, . 1/n? (1) n (2) 2n (3) 2/(n 1) (4) n(n 1)/2?
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What is H. M. of 1, 1/2, 1/3, . 1/n? (1) n (2) 2n (3) 2/(n 1) (4) n(n 1)/2? for CA Foundation 2025 is part of CA Foundation preparation. The Question and answers have been prepared according to the CA Foundation exam syllabus. Information about What is H. M. of 1, 1/2, 1/3, . 1/n? (1) n (2) 2n (3) 2/(n 1) (4) n(n 1)/2? covers all topics & solutions for CA Foundation 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for What is H. M. of 1, 1/2, 1/3, . 1/n? (1) n (2) 2n (3) 2/(n 1) (4) n(n 1)/2?.
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