The sum of four numbers in A.P. is 4 and their product is 385. Find the numbers.

Answer

To solve the problem of finding four numbers in Arithmetic Progression (A.P.) whose sum is 4 and product is 385, we can follow these steps:

### Step 1: Define the Numbers
Let the four numbers in A.P. be represented as:
- \( a - 3d \)
- \( a - d \)
- \( a + d \)
- \( a + 3d \)

### Step 2: Set Up the Sum Equation
According to the problem, the sum of these numbers is 4:
\[ (a - 3d) + (a - d) + (a + d) + (a + 3d) = 4 \]
Simplifying this, we get:
\[ 4a = 4 \]
Thus, we find:
\[ a = 1 \]

### Step 3: Substitute \( a \) Back
Now, substituting \( a \) back into our expressions for the numbers, we have:
- \( 1 - 3d \)
- \( 1 - d \)
- \( 1 + d \)
- \( 1 + 3d \)

### Step 4: Set Up the Product Equation
Next, we know that the product of these numbers is 385:
\[ (1 - 3d)(1 - d)(1 + d)(1 + 3d) = 385 \]
Using the difference of squares, we can rewrite this as:
\[ (1 - 9d^2)(1 - d^2) = 385 \]

### Step 5: Expand the Product
Expanding this gives:
\[ 1 - d^2 - 9d^2 + 9d^4 = 385 \]
This simplifies to:
\[ 9d^4 - 10d^2 + 1 - 385 = 0 \]
or:
\[ 9d^4 - 10d^2 - 384 = 0 \]

### Step 6: Let \( x = d^2 \)
Let \( x = d^2 \). The equation becomes:
\[ 9x^2 - 10x - 384 = 0 \]

### Step 7: Solve the Quadratic Equation
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 9, b = -10, c = -384 \):
\[ x = \frac{10 \pm \sqrt{(-10)^2 - 4 \cdot 9 \cdot (-384)}}{2 \cdot 9} \]
Calculating the discriminant:
\[ x = \frac{10 \pm \sqrt{100 + 13824}}{18} \]
\[ x = \frac{10 \pm \sqrt{13924}}{18} \]
Calculating \( \sqrt{13924} \):
\[ \sqrt{13924} = 118.2 \quad (\text{approximately}) \]
Thus:
\[ x = \frac{10 \pm 118.2}{18} \]
Calculating the two possible values for \( x \):
1. \( x_1 = \frac{128.2}{18} \approx 7.13 \)
2. \( x_2 = \frac{-108.2}{18} \) (not valid since \( x \) must be non-negative)

### Step 8: Find \( d \)
Now, since \( x = d^2 \):
\[ d^2 \approx 7.13 \implies d \approx \pm \sqrt{7.13} \approx \pm 2.67 \]

### Step 9: Calculate the Numbers
Now substituting back to find the numbers:
1. For \( d = 2.67 \):
- \( 1 - 3(2.67) \approx -6.01 \)
- \( 1 - 2.67 \approx -1.67 \)
- \( 1 + 2.67 \approx 3.67 \)
- \( 1 + 3(2.67) \approx 9.01 \)

2. For \( d = -2.67 \):
- \( 1 - 3(-2.67) \approx 9.01 \)
- \( 1 - (-2.67) \approx 3.67 \)
- \( 1 + (-2.67) \approx -1.67 \)
- \( 1 + 3(-2.67) \approx -6.01 \)

### Final Result
Thus, the four numbers in A.P. can be:
- \( -6, -\frac{5}{3}, \frac{11}{3}, 9 \) or
- \( 9, \frac{11}{3}, -\frac{5}{3}, -6 \)