Determine the 4th and 8th term of the harmonic progression 6, 4, 3,

A **harmonic progression (HP)** is a sequence of terms whose reciprocals form an arithmetic progression (AP). Given the harmonic progression \(6, 4, 3, \dots\), let's determine the 4th and 8th terms.

### Step 1: Find the arithmetic progression (AP)
The given HP is: \(6, 4, 3, \dots\). First, convert these terms to their reciprocals to form an AP:
\[
\frac{1}{6}, \frac{1}{4}, \frac{1}{3}, \dots
\]
We check if these terms form an AP by calculating the common difference:
\[
\text{Difference} = \frac{1}{4} - \frac{1}{6} = \frac{3 - 2}{12} = \frac{1}{12}
\]
Thus, the AP has a common difference of \(\frac{1}{12}\).

### Step 2: General term of the AP
The general term of an AP is given by:
\[
a_n = a_1 + (n-1) \cdot d
\]
where \(a_1\) is the first term and \(d\) is the common difference. For this AP:
\[
a_1 = \frac{1}{6}, \quad d = \frac{1}{12}
\]
Thus, the general term becomes:
\[
a_n = \frac{1}{6} + (n-1) \cdot \frac{1}{12}
\]

### Step 3: Find the 4th term of the AP
Substitute \(n = 4\) into the general term formula:
\[
a_4 = \frac{1}{6} + (4-1) \cdot \frac{1}{12} = \frac{1}{6} + \frac{3}{12} = \frac{1}{6} + \frac{1}{4}
\]
To add these, find a common denominator:
\[
a_4 = \frac{2}{12} + \frac{3}{12} = \frac{5}{12}
\]
The reciprocal of \(\frac{5}{12}\) gives the 4th term of the HP:
\[
\text{4th term of HP} = \frac{12}{5}
\]

### Step 4: Find the 8th term of the AP
Substitute \(n = 8\) into the general term formula:
\[
a_8 = \frac{1}{6} + (8-1) \cdot \frac{1}{12} = \frac{1}{6} + \frac{7}{12} = \frac{2}{12} + \frac{7}{12} = \frac{9}{12} = \frac{3}{4}
\]
The reciprocal of \(\frac{3}{4}\) gives the 8th term of the HP:
\[
\text{8th term of HP} = \frac{4}{3}
\]

### Final Answer:
- The 4th term of the harmonic progression is \(\frac{12}{5}\).
- The 8th term of the harmonic progression is \(\frac{4}{3}\).

Would you like more details or have any further questions?

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Here are 5 related questions to expand on the topic:
1. How do you derive the formula for the general term of an arithmetic progression?
2. What is the significance of harmonic progressions in physics or engineering?
3. How would you calculate the sum of the first \(n\) terms of a harmonic progression?
4. How does the behavior of harmonic progressions differ from geometric progressions?
5. Can harmonic progressions have negative terms? What would that imply?

**Tip:** When working with harmonic progressions, always convert the sequence to an arithmetic progression first by taking the reciprocals of the terms!

Determine the 4th and 8th term of the harmonic progression 6, 4, 3

Learn how to find the 4th and 8th terms of a harmonic progression with initial terms 6, 4, 3. This problem involves concepts of harmonic progression, arithmetic progression, and reciprocal relationships. Step-by-step solution included, suitable for Grades 9-11.

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