The sum of 3 numbers in A.P. is 15. If we add 1, 4, 19 respectively, then the new numbers form a G.P. Find the numbers of the A.P.

Explanation

To solve the problem, we need to find three numbers in Arithmetic Progression (A.P.) whose sum is 15, and when we add 1, 4, and 19 to these numbers respectively, the new numbers form a Geometric Progression (G.P.).

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

### Step 2: Set up the equation for the sum of the numbers.
According to the problem, the sum of these three numbers is 15:
\[ (a - d) + a + (a + d) = 15 \]
Simplifying this:
\[ 3a = 15 \]
Thus, we can solve for \( a \):
\[ a = 5 \]

### Step 3: Substitute \( a \) back into the expressions for the A.P. numbers.
Now substituting \( a = 5 \) into the expressions for the A.P. numbers, we get:
- First number: \( 5 - d \)
- Second number: \( 5 \)
- Third number: \( 5 + d \)

### Step 4: Add the respective values to form a G.P.
Now, we add 1, 4, and 19 to the respective numbers:
- New first number: \( (5 - d) + 1 = 6 - d \)
- New second number: \( 5 + 4 = 9 \)
- New third number: \( (5 + d) + 19 = 24 + d \)

### Step 5: Set up the condition for G.P.
For these new numbers to be in G.P., the square of the middle term must equal the product of the other two terms:
\[ 9^2 = (6 - d)(24 + d) \]
Calculating \( 9^2 \):
\[ 81 = (6 - d)(24 + d) \]

### Step 6: Expand and simplify the equation.
Expanding the right side:
\[ 81 = 144 + 6d - 24d - d^2 \]
This simplifies to:
\[ 81 = 144 - 18d - d^2 \]

### Step 7: Rearrange the equation.
Rearranging gives:
\[ d^2 + 18d + (81 - 144) = 0 \]
This simplifies to:
\[ d^2 + 18d - 63 = 0 \]

### Step 8: Factor the quadratic equation.
Now we factor the quadratic:
\[ (d + 21)(d - 3) = 0 \]
Thus, the solutions for \( d \) are:
\[ d = -21 \quad \text{or} \quad d = 3 \]

### Step 9: Find the A.P. numbers for both values of \( d \).
1. If \( d = -21 \):
- First number: \( 5 - (-21) = 5 + 21 = 26 \)
- Second number: \( 5 \)
- Third number: \( 5 + (-21) = 5 - 21 = -16 \)

So the numbers are \( 26, 5, -16 \).

2. If \( d = 3 \):
- First number: \( 5 - 3 = 2 \)
- Second number: \( 5 \)
- Third number: \( 5 + 3 = 8 \)

So the numbers are \( 2, 5, 8 \).

### Conclusion:
The two sets of numbers in A.P. that satisfy the conditions of the problem are:
1. \( 26, 5, -16 \)
2. \( 2, 5, 8 \)