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

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

Options:

Correct Answer: 1/6

Solution:

The reciprocals are 1, 2, and 3, which are in arithmetic progression. The next term in the sequence of reciprocals is 4, so the fourth term is 1/4.

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

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

Step 1: Identify the first three terms of the harmonic progression, which are 1, 1/2, and 1/3.
Step 2: Find the reciprocals of these terms. The reciprocal of 1 is 1, the reciprocal of 1/2 is 2, and the reciprocal of 1/3 is 3.
Step 3: Now we have the sequence 1, 2, 3 from the reciprocals. This sequence is an arithmetic progression (AP) because the difference between consecutive terms is constant (2 - 1 = 1 and 3 - 2 = 1).
Step 4: To find the next term in this arithmetic progression, add 1 to the last term (3). So, 3 + 1 = 4.
Step 5: The fourth term in the harmonic progression is the reciprocal of the new term we found in the AP. The reciprocal of 4 is 1/4.
Step 6: Therefore, the fourth term of the harmonic progression is 1/4.

Harmonic Progression – A sequence of numbers is in harmonic progression if the reciprocals of the numbers 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 find terms in sequences by using their reciprocals.