To insert 4 Harmonic Means (H.M.) between \( \frac{3}{4} \) and \( \frac{3}{19} \), we will follow these steps:
### Step 1: Understand the relationship between Harmonic Means and Arithmetic Progression
The Harmonic Means between two numbers can be derived from the corresponding Arithmetic Progression (A.P.) of their reciprocals. If we denote the Harmonic Means as \( H_1, H_2, H_3, H_4 \), we can express them as:
\[
\frac{1}{H_1}, \frac{1}{H_2}, \frac{1}{H_3}, \frac{1}{H_4}
\]
These values will form an A.P. with the first term \( \frac{1}{\frac{3}{4}} = \frac{4}{3} \) and the last term \( \frac{1}{\frac{3}{19}} = \frac{19}{3} \).
### Step 2: Find the common difference of the A.P.
Let the common difference of the A.P. be \( d \). The terms of the A.P. can be expressed as:
- First term: \( a = \frac{4}{3} \)
- Second term: \( a + d \)
- Third term: \( a + 2d \)
- Fourth term: \( a + 3d \)
- Fifth term: \( a + 4d = \frac{19}{3} \)
From this, we can set up the equation:
\[
\frac{4}{3} + 4d = \frac{19}{3}
\]
### Step 3: Solve for \( d \)
Rearranging the equation gives:
\[
4d = \frac{19}{3} - \frac{4}{3} = \frac{15}{3} = 5
\]
Thus,
\[
d = \frac{5}{4}
\]
### Step 4: Calculate the terms of the A.P.
Now we can find the terms of the A.P.:
- \( \frac{1}{H_1} = a = \frac{4}{3} \)
- \( \frac{1}{H_2} = a + d = \frac{4}{3} + \frac{5}{4} \)
To add these fractions, we need a common denominator:
\[
\frac{4}{3} = \frac{16}{12}, \quad \frac{5}{4} = \frac{15}{12} \quad \Rightarrow \quad \frac{1}{H_2} = \frac{16}{12} + \frac{15}{12} = \frac{31}{12}
\]
- \( \frac{1}{H_3} = a + 2d = \frac{4}{3} + 2 \cdot \frac{5}{4} = \frac{4}{3} + \frac{10}{4} \)
Converting \( \frac{4}{3} \) to a common denominator:
\[
\frac{4}{3} = \frac{16}{12}, \quad \frac{10}{4} = \frac{30}{12} \quad \Rightarrow \quad \frac{1}{H_3} = \frac{16}{12} + \frac{30}{12} = \frac{46}{12}
\]
- \( \frac{1}{H_4} = a + 3d = \frac{4}{3} + 3 \cdot \frac{5}{4} = \frac{4}{3} + \frac{15}{4} \)
Converting \( \frac{4}{3} \) to a common denominator:
\[
\frac{4}{3} = \frac{16}{12}, \quad \frac{15}{4} = \frac{45}{12} \quad \Rightarrow \quad \frac{1}{H_4} = \frac{16}{12} + \frac{45}{12} = \frac{61}{12}
\]
### Step 5: Calculate the Harmonic Means
Now we can find the values of \( H_1, H_2, H_3, H_4 \):
- \( H_1 = \frac{3}{4} \)
- \( H_2 = \frac{12}{31} \)
- \( H_3 = \frac{12}{46} = \frac{6}{23} \)
- \( H_4 = \frac{12}{61} \)
### Final Answer
Thus, the 4 Harmonic Means between \( \frac{3}{4} \) and \( \frac{3}{19} \) are:
\[
H_1 = \frac{3}{4}, \quad H_2 = \frac{12}{31}, \quad H_3 = \frac{6}{23}, \quad H_4 = \frac{12}{61}
\]
