Identify the term that does not belong to the harmonic progression: 1/2, 1/3, 1/

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Question: Identify the term that does not belong to the harmonic progression: 1/2, 1/3, 1/4, 1/5, 1/6.

Options:

Correct Answer: 1/4

Solution:

The terms 1/2, 1/3, 1/4, and 1/5 are in harmonic progression, but 1/6 does not fit as its reciprocal does not maintain the arithmetic progression.

Identify the term that does not belong to the harmonic progression: 1/2, 1/3, 1/4, 1/5, 1/6.

Identify the term that does not belong to the harmonic progression: 1/2, 1/3, 1/4, 1/5, 1/6.

Step 1: Understand what a harmonic progression is. A harmonic progression is a sequence of numbers where the reciprocals of the numbers form an arithmetic progression.
Step 2: Find the reciprocals of the given terms: 1/2 becomes 2, 1/3 becomes 3, 1/4 becomes 4, 1/5 becomes 5, and 1/6 becomes 6.
Step 3: List the reciprocals: 2, 3, 4, 5, 6.
Step 4: Check if these numbers (2, 3, 4, 5, 6) form an arithmetic progression. An arithmetic progression has a constant difference between consecutive terms.
Step 5: Calculate the differences: 3 - 2 = 1, 4 - 3 = 1, 5 - 4 = 1, but 6 - 5 = 1. All differences are equal, so 2, 3, 4, 5 are in arithmetic progression.
Step 6: Since the first four terms (1/2, 1/3, 1/4, 1/5) have their reciprocals in arithmetic progression, but 1/6 does not fit, we conclude that 1/6 does not belong to the harmonic progression.

Harmonic Progression – A sequence of numbers is in harmonic progression if the reciprocals of the terms form an arithmetic progression.
Reciprocal Relationship – Understanding how the reciprocals of the terms relate to each other is crucial for identifying harmonic progressions.
Arithmetic Progression – A sequence of numbers is in arithmetic progression if the difference between consecutive terms is constant.