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If the terms of a harmonic progression are 3, 6, and x, what is the value of x?

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Question: If the terms of a harmonic progression are 3, 6, and x, what is the value of x?

Options:

  1. 9
  2. 12
  3. 15
  4. 18

Correct Answer: 12

Solution: 

The reciprocals of the terms are 1/3, 1/6, and 1/x. Since they form an arithmetic progression, we can set up the equation: 1/6 - 1/3 = 1/x - 1/6, solving gives x = 12.

If the terms of a harmonic progression are 3, 6, and x, what is the value of x?

Practice Questions

Q1
If the terms of a harmonic progression are 3, 6, and x, what is the value of x?
  1. 9
  2. 12
  3. 15
  4. 18

Questions & Step-by-Step Solutions

If the terms of a harmonic progression are 3, 6, and x, what is the value of x?
  • Step 1: Identify the terms of the harmonic progression, which are 3, 6, and x.
  • Step 2: Write down the reciprocals of these terms: 1/3, 1/6, and 1/x.
  • Step 3: Recognize that the reciprocals form an arithmetic progression (AP).
  • Step 4: In an AP, the difference between consecutive terms is constant. So, set up the equation: (1/6) - (1/3) = (1/x) - (1/6).
  • Step 5: Calculate (1/6) - (1/3). To do this, find a common denominator (which is 6): (1/6) - (2/6) = -1/6.
  • Step 6: Now, rewrite the equation: -1/6 = (1/x) - (1/6).
  • Step 7: Add (1/6) to both sides to isolate (1/x): (1/x) = -1/6 + 1/6 = 0.
  • Step 8: Since we made a mistake in the previous steps, let's correct it: Instead, we should have (1/6) - (1/3) = (1/x) - (1/6) gives us (1/x) = (1/6) + (1/6) = 2/6 = 1/3.
  • Step 9: Now, take the reciprocal of both sides: x = 3.
  • Harmonic Progression – A sequence of numbers is in harmonic progression if the reciprocals of the terms form an arithmetic progression.
  • Arithmetic Progression – A sequence of numbers is in arithmetic progression if the difference between consecutive terms is constant.
  • Reciprocal Relationships – Understanding how to manipulate and relate the reciprocals of numbers in sequences.
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