To solve the problem step by step, we need to calculate the determinant \( D = |a_1, a_2, a_3, 5, 4, a_6, a_7, a_8, a_9| \) given that \( a_1, a_2, a_3, a_4, a_5, a_6, a_7, a_8, a_9 \) are in Harmonic Progression (H.P.).
### Step 1: Understanding Harmonic Progression
In H.P., the reciprocals of the terms are in Arithmetic Progression (A.P.). Thus, if \( a_n \) is the \( n \)-th term in H.P., we can express it as:
\[
a_n = \frac{1}{b_n}
\]
where \( b_n \) is the \( n \)-th term in A.P.
### Step 2: Finding the Terms
Given that \( a_4 = 5 \) and \( a_5 = 4 \), we can find the terms in A.P.:
\[
b_4 = \frac{1}{5}, \quad b_5 = \frac{1}{4}
\]
The common difference \( d \) of the A.P. can be calculated as:
\[
d = b_5 - b_4 = \frac{1}{4} - \frac{1}{5} = \frac{5 - 4}{20} = \frac{1}{20}
\]
Now, we can express the terms \( b_n \) as:
\[
b_n = b_1 + (n-1)d
\]
Substituting \( d \):
\[
b_n = b_1 + \frac{(n-1)}{20}
\]
### Step 3: Finding \( a_n \)
We can express \( a_n \) in terms of \( b_n \):
\[
a_n = \frac{1}{b_n} = \frac{1}{b_1 + \frac{(n-1)}{20}}
\]
To find \( b_1 \), we can use the values of \( b_4 \) and \( b_5 \):
\[
b_4 = b_1 + \frac{3}{20} = \frac{1}{5} \implies b_1 = \frac{1}{5} - \frac{3}{20} = \frac{4 - 3}{20} = \frac{1}{20}
\]
Thus, we can express all terms:
\[
b_n = \frac{1}{20} + \frac{(n-1)}{20} = \frac{n}{20}
\]
This gives us:
\[
a_n = \frac{20}{n}
\]
### Step 4: Substitute the Values into the Determinant
Now substituting the values of \( a_n \) into the determinant:
\[
D = \left| \begin{array}{ccccccccc}
20 & 20/2 & 20/3 & 5 & 4 & 20/6 & 20/7 & 20/8 & 20/9
\end{array} \right|
\]
### Step 5: Factor Out Common Terms
We can factor out \( 20 \) from each row:
\[
D = 20^3 \cdot \left| \begin{array}{ccccccccc}
1 & 1/2 & 1/3 & 1 & 4/5 & 1/6 & 1/7 & 1/8 & 1/9
\end{array} \right|
\]
### Step 6: Simplifying the Determinant
To simplify the determinant, we can perform row operations:
- \( R_1 \to R_1 - R_2 \)
- \( R_2 \to R_2 - R_3 \)
This will help us simplify the determinant further.
### Step 7: Calculate the Determinant
After performing the necessary row operations and calculating the determinant, we find:
\[
D = \frac{50}{21}
\]
### Step 8: Greatest Integer Function
Finally, we apply the greatest integer function:
\[
\lfloor D \rfloor = \lfloor \frac{50}{21} \rfloor = 2
\]
### Final Answer
Thus, the value of \( [D] \) is:
\[
\boxed{2}
\]
