Find 4 numbers in A.P. whose sum is 50 and greatest number is 4 times the smallest number.

Explanation

To find 4 numbers in Arithmetic Progression (A.P.) whose sum is 50 and where the greatest number is 4 times the smallest number, we can follow these steps:

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

### Step 2: Set up the equation for the sum of the numbers.
The sum of these four numbers is given as 50:
\[ (a - 3d) + (a - d) + (a + d) + (a + 3d) = 50 \]

### Step 3: Simplify the equation.
Combine like terms:
\[ 4a = 50 \]

### Step 4: Solve for \( a \).
Dividing both sides by 4:
\[ a = \frac{50}{4} = 12.5 \]

### Step 5: Set up the equation based on the condition of the greatest and smallest number.
According to the problem, the greatest number is 4 times the smallest number:
\[ a + 3d = 4(a - 3d) \]

### Step 6: Expand and simplify the equation.
Expanding the right side:
\[ a + 3d = 4a - 12d \]
Rearranging gives:
\[ 3d + 12d = 4a - a \]
\[ 15d = 3a \]

### Step 7: Substitute the value of \( a \) into the equation.
Substituting \( a = 12.5 \):
\[ 15d = 3 \times 12.5 \]
\[ 15d = 37.5 \]

### Step 8: Solve for \( d \).
Dividing both sides by 15:
\[ d = \frac{37.5}{15} = 2.5 \]

### Step 9: Find the four numbers.
Now substitute \( a \) and \( d \) back into the expressions for the four numbers:
1. \( a - 3d = 12.5 - 3(2.5) = 12.5 - 7.5 = 5 \)
2. \( a - d = 12.5 - 2.5 = 10 \)
3. \( a + d = 12.5 + 2.5 = 15 \)
4. \( a + 3d = 12.5 + 3(2.5) = 12.5 + 7.5 = 20 \)

### Conclusion:
The four numbers in A.P. are:
\[ 5, 10, 15, 20 \]