Let a,x,y,z,b be in AP, where x+y+z=15. let a,p,q,r,b be in HP, where `p^(-1)+q^(-1)+r^(-1) =(5)/(3)`
Q. What is the value of pqr ?

Options

A. `(35)/(243)`
B. `(81)/(35)`
C. `(243)/(35)`
D. Cannot be determined

Answer

To solve the problem step by step, we will first analyze the information given about the arithmetic progression (AP) and harmonic progression (HP).

### Step 1: Understand the AP condition
Given that \( a, x, y, z, b \) are in AP, we know that:
- The average of \( x \) and \( y \) is equal to the average of \( y \) and \( z \).
- This can be expressed as:
\[ x + z = 2y \]

### Step 2: Use the sum condition
We are given that:
\[ x + y + z = 15 \]
From the AP condition, we can substitute \( x + z \) with \( 2y \):
\[ 2y + y = 15 \implies 3y = 15 \implies y = 5 \]

### Step 3: Find \( x \) and \( z \)
Now that we have \( y = 5 \), we can substitute back to find \( x \) and \( z \):
\[ x + z = 2y = 2 \times 5 = 10 \]
Thus, we can express \( z \) in terms of \( x \):
\[ z = 10 - x \]

### Step 4: Understand the HP condition
Next, we know that \( a, p, q, r, b \) are in HP. This means that the reciprocals \( \frac{1}{a}, \frac{1}{p}, \frac{1}{q}, \frac{1}{r}, \frac{1}{b} \) are in AP. Therefore:
\[ \frac{1}{a} + \frac{1}{b} = \frac{2}{p} + \frac{1}{q} + \frac{1}{r} \]

### Step 5: Use the given HP condition
We are given:
\[ \frac{1}{p} + \frac{1}{q} + \frac{1}{r} = \frac{5}{3} \]
Let’s denote:
\[ \frac{1}{p} + \frac{1}{q} = S \quad \text{and} \quad \frac{1}{r} = T \]
From the HP condition, we can express:
\[ S + T = \frac{5}{3} \]
Also, since \( \frac{1}{p} + \frac{1}{q} = \frac{2}{q} \) (from the property of HP), we can write:
\[ \frac{2}{q} + \frac{1}{r} = \frac{5}{3} \]

### Step 6: Solve for \( q \) and \( r \)
Let’s express \( \frac{1}{r} \) in terms of \( q \):
\[ \frac{1}{r} = \frac{5}{3} - \frac{2}{q} \]

### Step 7: Find \( pqr \)
Now, we can find \( pqr \) using the relationships we have established. We know:
\[ \frac{1}{p} + \frac{1}{q} + \frac{1}{r} = \frac{5}{3} \]
Let’s denote \( pqr = k \). The product of the reciprocals gives:
\[ \frac{1}{pqr} = \frac{1}{p} + \frac{1}{q} + \frac{1}{r} \]
Thus:
\[ pqr = \frac{1}{\frac{5}{3}} = \frac{3}{5} \]

### Final Result
To find \( pqr \), we can multiply the values we have:
\[ pqr = 9 \times 3 \times 5 \times 7 = 945 \]

### Conclusion
The value of \( pqr \) is \( 945 \).