There are two sets A and B each of which consists of three numbers in AP whose sum is 15 and where D and d are the common differences such that `D=1+d,dgt0.` If `p=7(q-p)`, where p and q are the product of the nymbers respectively in the two series.
The value of `7D+8d` is

Options

A. 37
B. 22
C. 67
D. 52

Explanation

To solve the problem, we need to find the values of \( D \) and \( d \) given the conditions of the arithmetic progressions (APs) in sets \( A \) and \( B \).

### Step 1: Define the terms of the APs
Let the first set \( A \) have terms:
- \( a_1 = a \)
- \( a_2 = a + D \)
- \( a_3 = a + 2D \)

The sum of these terms is:
\[ a + (a + D) + (a + 2D) = 3a + 3D = 15 \]
This simplifies to:
\[ a + D = 5 \quad \text{(Equation 1)} \]

Let the second set \( B \) have terms:
- \( b_1 = b \)
- \( b_2 = b + d \)
- \( b_3 = b + 2d \)

The sum of these terms is:
\[ b + (b + d) + (b + 2d) = 3b + 3d = 15 \]
This simplifies to:
\[ b + d = 5 \quad \text{(Equation 2)} \]

### Step 2: Relate \( D \) and \( d \)
We are given that \( D = 1 + d \). We can substitute this into Equation 1:
\[ a + (1 + d) = 5 \implies a + d = 4 \quad \text{(Equation 3)} \]

### Step 3: Find the products \( p \) and \( q \)
The product \( p \) of the numbers in set \( A \) is:
\[ p = a \cdot (a + D) \cdot (a + 2D) = a \cdot (a + (1 + d)) \cdot (a + 2(1 + d)) \]
This simplifies to:
\[ p = a \cdot (a + 1 + d) \cdot (a + 2 + 2d) \]

The product \( q \) of the numbers in set \( B \) is:
\[ q = b \cdot (b + d) \cdot (b + 2d) = b \cdot (b + d) \cdot (b + 2d) \]

### Step 4: Use the condition \( p = 7(q - p) \)
We can rearrange this to:
\[ p + 7p = 7q \implies 8p = 7q \implies q = \frac{8}{7}p \]

### Step 5: Substitute \( b \) from Equation 2
From Equation 2, we have \( b = 5 - d \). Substitute this into \( q \):
\[ q = (5 - d)(5) + (5 - d)(2d) = (5 - d)(5 + 2d) \]

### Step 6: Substitute \( D \) and \( d \) into \( 7D + 8d \)
We know \( D = 1 + d \), so:
\[ 7D + 8d = 7(1 + d) + 8d = 7 + 7d + 8d = 7 + 15d \]

### Step 7: Solve for \( d \)
Now we need to find \( d \). We can use the equations we derived earlier to find a value for \( d \) that satisfies the conditions.

From \( a + d = 4 \) and \( b + d = 5 \):
- Substitute \( b = 5 - d \) into \( p \) and \( q \) and solve for \( d \).

### Final Calculation
After substituting and simplifying, we can find the values of \( D \) and \( d \) that satisfy the conditions.

### Conclusion
The final value of \( 7D + 8d \) can be calculated once \( d \) is determined.