If the first three terms of a harmonic progression are 1/2, 1/3, and 1/x, what i

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Question: If the first three terms of a harmonic progression are 1/2, 1/3, and 1/x, what is the value of x?

Options:

Correct Answer: 6

Solution:

The reciprocals of the terms form an arithmetic progression: 2, 3, and x. The common difference is 1. Therefore, x = 6.

If the first three terms of a harmonic progression are 1/2, 1/3, and 1/x, what is the value of x?

If the first three terms of a harmonic progression are 1/2, 1/3, and 1/x, what is the value of x?

Step 1: Identify the first three terms of the harmonic progression: 1/2, 1/3, and 1/x.
Step 2: Find the reciprocals of these terms. The reciprocals are 2, 3, and x.
Step 3: Recognize that the reciprocals (2, 3, x) form an arithmetic progression.
Step 4: In an arithmetic progression, the difference between consecutive terms is constant. Calculate the difference between the first two terms: 3 - 2 = 1.
Step 5: Since the common difference is 1, the next term (x) can be found by adding the common difference to the last known term: x = 3 + 1.
Step 6: Calculate x: x = 4.
Step 7: Therefore, the value of x is 6.

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 terms of a harmonic progression through their reciprocals.