To determine which of the following statements is not true when positive quantities \( a, b, c \) are in Harmonic Progression (H.P.), we will analyze the properties of H.P. and the relationships between the Harmonic Mean (H.M.), Geometric Mean (G.M.), and Arithmetic Mean (A.M.).
### Step-by-Step Solution:
1. **Understanding H.P.**:
If \( a, b, c \) are in H.P., then their reciprocals \( \frac{1}{a}, \frac{1}{b}, \frac{1}{c} \) are in Arithmetic Progression (A.P.). This means:
\[
2b = a + c
\]
2. **Finding the Harmonic Mean**:
The Harmonic Mean (H.M.) of \( a, b, c \) can be calculated as:
\[
H.M. = \frac{3}{\frac{1}{a} + \frac{1}{b} + \frac{1}{c}} = \frac{3abc}{ab + ac + bc}
\]
3. **Finding the Geometric Mean**:
The Geometric Mean (G.M.) of \( a, b, c \) is given by:
\[
G.M. = \sqrt[3]{abc}
\]
4. **Finding the Arithmetic Mean**:
The Arithmetic Mean (A.M.) of \( a, b, c \) is:
\[
A.M. = \frac{a + b + c}{3}
\]
5. **Inequalities Involving Means**:
We know from the properties of means that:
\[
H.M. \leq G.M. \leq A.M.
\]
Given that \( b = \frac{a+c}{2} \) (from the H.P. condition), we can substitute this into the inequality:
\[
b \leq \sqrt{ac} \quad \text{and} \quad \frac{a+c}{2} \leq \frac{a+b+c}{3}
\]
6. **Analyzing the Options**:
We need to check which of the following statements is not true based on the above inequalities:
- \( b < \frac{a+c}{2} \)
- \( b < \sqrt{ac} \)
- \( b > \frac{a+c}{2} \)
- \( b < \frac{a+b+c}{3} \)
From our analysis, we see that \( b \) cannot be greater than \( \frac{a+c}{2} \) since \( b \) is defined as \( \frac{a+c}{2} \). Therefore, the statement \( b > \frac{a+c}{2} \) is not true.
### Conclusion:
Thus, the statement that is not true is:
\[
\text{Option: } b > \frac{a+c}{2}
\]
