In a harmonic progression, if the first term is 1 and the second term is 1/2, wh

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Question: In a harmonic progression, if the first term is 1 and the second term is 1/2, what is the third term?

Options:

Correct Answer: 1/3

Solution:

The reciprocals are 1 and 2, which are in arithmetic progression. The third term\'s reciprocal is 3, so the third term is 1/3.

In a harmonic progression, if the first term is 1 and the second term is 1/2, what is the third term?

In a harmonic progression, if the first term is 1 and the second term is 1/2, what is the third term?

Step 1: Understand that a harmonic progression (HP) is a sequence of numbers whose reciprocals form an arithmetic progression (AP).
Step 2: Identify the first term of the HP, which is 1. The reciprocal of 1 is 1.
Step 3: Identify the second term of the HP, which is 1/2. The reciprocal of 1/2 is 2.
Step 4: Now we have the first two terms of the AP: 1 and 2.
Step 5: In an arithmetic progression, the difference between consecutive terms is constant. The difference between 1 and 2 is 1.
Step 6: To find the third term of the AP, add the common difference (1) to the second term (2). So, 2 + 1 = 3.
Step 7: The third term of the HP is the reciprocal of the third term of the AP. The reciprocal of 3 is 1/3.
Step 8: Therefore, the third term of the harmonic progression is 1/3.

Harmonic Progression – A sequence of numbers is in harmonic progression if the reciprocals of the terms form an arithmetic progression.
Reciprocal Relationship – Understanding how to find the reciprocal of terms in a harmonic progression to derive the next term.
Arithmetic Progression – Recognizing that the reciprocals of the terms in a harmonic progression must form an arithmetic progression.