If the terms of a harmonic progression are 4, 2, and x, what is the value of x?

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Question: If the terms of a harmonic progression are 4, 2, and x, what is the value of x?

Options:

Correct Answer: 3

Solution:

The reciprocals are 1/4, 1/2, and 1/x. The common difference is 1/2 - 1/4 = 1/4, so 1/x = 1/2 + 1/4 = 3/4, thus x = 4/3.

If the terms of a harmonic progression are 4, 2, and x, what is the value of x?

If the terms of a harmonic progression are 4, 2, and x, what is the value of x?

Step 1: Understand that a harmonic progression (HP) is a sequence where the reciprocals of the terms form an arithmetic progression (AP).
Step 2: Write down the given terms of the HP: 4, 2, and x.
Step 3: Find the reciprocals of the terms: 1/4, 1/2, and 1/x.
Step 4: Identify the common difference in the AP formed by the reciprocals. Calculate the difference between the first two terms: 1/2 - 1/4.
Step 5: Simplify the difference: 1/2 - 1/4 = 1/4.
Step 6: Use the common difference to find the next term in the AP. Add the common difference (1/4) to the second term (1/2): 1/2 + 1/4.
Step 7: Simplify the result: 1/2 + 1/4 = 3/4.
Step 8: Set the third term (1/x) equal to the result from Step 7: 1/x = 3/4.
Step 9: Solve for x by taking the reciprocal of both sides: x = 4/3.

Harmonic Progression – A sequence of numbers is in harmonic progression if the reciprocals of the terms form an arithmetic progression.
Reciprocal Relationships – Understanding how to manipulate and relate the reciprocals of the terms in a harmonic progression.
Common Difference – Identifying and calculating the common difference in the arithmetic progression formed by the reciprocals.