Divide 56 in four parts in A.P. such that the ratio of the product of their extremes (1st and 4th) to the product of means (2nd and 3rd) is 5:6.

Answer

To solve the problem of dividing 56 into four parts in Arithmetic Progression (A.P.) such that the ratio of the product of the extremes (1st and 4th) to the product of the means (2nd and 3rd) is 5:6, we can follow these steps:

### Step 1: Define the terms in A.P.
Let the four parts in A.P. be:
- First term: \( a - 3d \)
- Second term: \( a - d \)
- Third term: \( a + d \)
- Fourth term: \( a + 3d \)

### Step 2: Set up the equation for the sum of the parts
Since the sum of these four parts is 56, we can write:
\[ (a - 3d) + (a - d) + (a + d) + (a + 3d) = 56 \]
Simplifying this, we get:
\[ 4a = 56 \]
Thus,
\[ a = \frac{56}{4} = 14 \]

### Step 3: Set up the equation for the ratio of products
We need to find the ratio of the product of the extremes (1st and 4th) to the product of the means (2nd and 3rd):
\[ \frac{(a - 3d)(a + 3d)}{(a - d)(a + d)} = \frac{5}{6} \]

### Step 4: Simplify the products
Calculating the products:
- Product of extremes:
\[ (a - 3d)(a + 3d) = a^2 - 9d^2 \]
- Product of means:
\[ (a - d)(a + d) = a^2 - d^2 \]
Thus, we can rewrite the ratio as:
\[ \frac{a^2 - 9d^2}{a^2 - d^2} = \frac{5}{6} \]

### Step 5: Cross-multiply and simplify
Cross-multiplying gives:
\[ 6(a^2 - 9d^2) = 5(a^2 - d^2) \]
Expanding both sides:
\[ 6a^2 - 54d^2 = 5a^2 - 5d^2 \]
Rearranging gives:
\[ 6a^2 - 5a^2 = 54d^2 - 5d^2 \]
Thus,
\[ a^2 = 49d^2 \]

### Step 6: Solve for \( d \)
Taking the square root of both sides:
\[ a = \pm 7d \]
Since we found \( a = 14 \), we have:
\[ 14 = 7d \quad \text{or} \quad 14 = -7d \]
From \( 14 = 7d \):
\[ d = 2 \]
From \( 14 = -7d \):
\[ d = -2 \]

### Step 7: Find the four parts
1. **For \( d = 2 \)**:
- First term: \( 14 - 3(2) = 8 \)
- Second term: \( 14 - 2 = 12 \)
- Third term: \( 14 + 2 = 16 \)
- Fourth term: \( 14 + 3(2) = 20 \)

Thus, the four parts are \( 8, 12, 16, 20 \).

2. **For \( d = -2 \)**:
- First term: \( 14 - 3(-2) = 20 \)
- Second term: \( 14 - (-2) = 16 \)
- Third term: \( 14 + (-2) = 12 \)
- Fourth term: \( 14 + 3(-2) = 8 \)

Thus, the four parts are \( 20, 16, 12, 8 \).

### Final Answer
The four numbers can be \( 8, 12, 16, 20 \) or \( 20, 16, 12, 8 \).

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