If the terms of a harmonic progression are 1, 1/4, and 1/9, what is the common d

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Question: If the terms of a harmonic progression are 1, 1/4, and 1/9, what is the common difference of the corresponding arithmetic progression?

Options:

Correct Answer: 1/12

Solution:

The reciprocals are 1, 4, and 9. The common difference is 4 - 1 = 3.

If the terms of a harmonic progression are 1, 1/4, and 1/9, what is the common difference of the corresponding arithmetic progression?

If the terms of a harmonic progression are 1, 1/4, and 1/9, what is the common difference of the corresponding arithmetic progression?

Step 1: Understand that a harmonic progression (HP) is a sequence of numbers whose reciprocals form an arithmetic progression (AP).
Step 2: Write down the terms of the harmonic progression: 1, 1/4, and 1/9.
Step 3: Find the reciprocals of these terms: The reciprocal of 1 is 1, the reciprocal of 1/4 is 4, and the reciprocal of 1/9 is 9.
Step 4: Now we have the sequence of reciprocals: 1, 4, and 9.
Step 5: Identify that these numbers (1, 4, 9) form an arithmetic progression.
Step 6: To find the common difference of the arithmetic progression, subtract the first term from the second term: 4 - 1 = 3.
Step 7: Therefore, the common difference of the corresponding arithmetic progression is 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 Relationship – Understanding the relationship between harmonic and arithmetic progressions through the use of reciprocals.