Insert 6 harmonic means between 3 and `(6)/(23)`

Explanation

To insert 6 harmonic means between 3 and \( \frac{6}{23} \), we can follow these steps:

### Step 1: Understand the relationship between Harmonic Means and Arithmetic Progression
Harmonic means (H) between two numbers can be found by taking the reciprocals of those numbers and forming an arithmetic progression (AP) with them. If we denote the harmonic means as \( H_1, H_2, H_3, H_4, H_5, H_6 \), we can express this as:
\[ 3, H_1, H_2, H_3, H_4, H_5, H_6, \frac{6}{23} \]

### Step 2: Convert to their reciprocals
The reciprocals will be:
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