Given Information:
The numbers \( a, b, c, d \) are in Harmonic Progression (HP).
The Arithmetic Mean (AM) of \( ab, bc, cd \) is 9.
Step 1: Understanding HP
If \( a, b, c, d \) are in Harmonic Progression, then their reciprocals are in Arithmetic Progression (AP).
So, we can assume: \[ \frac{1}{a}, \frac{1}{b}, \frac{1}{c}, \frac{1}{d} \] are in AP, which means: \[ 2 \cdot \frac{1}{b} = \frac{1}{a} + \frac{1}{c} \] \[ 2 \cdot \frac{1}{c} = \frac{1}{b} + \frac{1}{d} \] Step 2: Expressing the AM condition
We are given: \[ \frac{ab + bc + cd}{3} = 9 \] \[ ab + bc + cd = 27 \] Step 3: Expressing in terms of \( ad \)
We use the property that in an HP sequence, the product of the first and last terms is related to the middle terms.
From the property: \[ ad = bc \] Thus, replacing \( bc \) in the equation: \[ ab + ad + cd = 27 \] Since \( ad = bc \), we can rewrite: \[ ab + bc + cd = 27 \] which remains valid.
Step 4: Finding \( ad \)
If \( ad = bc \), we substitute: \[ ad + ad + ad = 27 \] \[ 3ad = 27 \] \[ ad = 9 \] Conclusion: The value of \( ad \) is: 9