If the first three terms of a harmonic progression are a, b, and c, which of the

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Question: If the first three terms of a harmonic progression are a, b, and c, which of the following is true?

Options:

Correct Answer: 1/a + 1/c = 2/b

Solution:

In a harmonic progression, the reciprocals of the terms form an arithmetic progression, which leads to the relationship 1/a + 1/c = 2/b.

If the first three terms of a harmonic progression are a, b, and c, which of the following is true?

If the first three terms of a harmonic progression are a, b, and c, which of the following is true?

Step 1: Understand what a harmonic progression (HP) is. In an HP, the reciprocals of the terms form an arithmetic progression (AP).
Step 2: Identify the first three terms of the harmonic progression as a, b, and c.
Step 3: Write down the reciprocals of these terms: 1/a, 1/b, and 1/c.
Step 4: Since these reciprocals form an arithmetic progression, we can use the property of AP: the middle term (1/b) is the average of the other two terms (1/a and 1/c).
Step 5: Set up the equation based on the AP property: 1/b = (1/a + 1/c) / 2.
Step 6: Multiply both sides of the equation by 2 to eliminate the fraction: 2/b = 1/a + 1/c.
Step 7: Rearrange the equation to get the final relationship: 1/a + 1/c = 2/b.

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 the reciprocals of terms relate to each other in harmonic and arithmetic progressions.