Three numbers are in A.P. and their sum is 15. If 1,4 and 19 be added to these numbers respectively the number are in G.P. Find the numbers .

Answer

To solve the problem step-by-step, we will follow the reasoning laid out in the video transcript.

### Step 1: Define the numbers in A.P.
Let the three numbers in Arithmetic Progression (A.P.) be:
- First number: \( A \)
- Second number: \( A + D \)
- Third number: \( A + 2D \)

### Step 2: Set up the equation for the sum of the numbers.
According to the problem, the sum of these three numbers is 15:
\[ A + (A + D) + (A + 2D) = 15 \]
This simplifies to:
\[ 3A + 3D = 15 \]
Dividing the entire equation by 3 gives:
\[ A + D = 5 \quad \text{(Equation 1)} \]

### Step 3: Set up the equation for the numbers in G.P.
When we add 1, 4, and 19 to the three numbers respectively, they become:
- First number: \( A + 1 \)
- Second number: \( A + D + 4 \)
- Third number: \( A + 2D + 19 \)

These three numbers must be in Geometric Progression (G.P.). For numbers to be in G.P., the square of the middle term must equal the product of the other two terms:
\[ (A + D + 4)^2 = (A + 1)(A + 2D + 19) \]

### Step 4: Substitute \( D \) from Equation 1.
From Equation 1, we have:
\[ D = 5 - A \]
Substituting \( D \) in the G.P. equation:
\[ (A + (5 - A) + 4)^2 = (A + 1)(A + 2(5 - A) + 19) \]
This simplifies to:
\[ (9)^2 = (A + 1)(A + 10 - 2A + 19) \]
\[ 81 = (A + 1)(29 - A) \]

### Step 5: Expand and rearrange the equation.
Expanding the right side:
\[ 81 = 29A - A^2 + 29 - A \]
This simplifies to:
\[ 81 = -A^2 + 28A + 29 \]
Rearranging gives:
\[ A^2 - 28A + 52 = 0 \]

### Step 6: Solve the quadratic equation.
Using the quadratic formula \( A = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 1, b = -28, c = 52 \):
\[ A = \frac{28 \pm \sqrt{(-28)^2 - 4 \cdot 1 \cdot 52}}{2 \cdot 1} \]
Calculating the discriminant:
\[ A = \frac{28 \pm \sqrt{784 - 208}}{2} \]
\[ A = \frac{28 \pm \sqrt{576}}{2} \]
\[ A = \frac{28 \pm 24}{2} \]
Calculating the two possible values for \( A \):
1. \( A = \frac{52}{2} = 26 \)
2. \( A = \frac{4}{2} = 2 \)

### Step 7: Find corresponding values of \( D \).
Using \( D = 5 - A \):
1. If \( A = 26 \):
\[ D = 5 - 26 = -21 \]
The numbers are:
- \( 26, 5, -16 \)

2. If \( A = 2 \):
\[ D = 5 - 2 = 3 \]
The numbers are:
- \( 2, 5, 8 \)

### Final Answer:
The original numbers can be either:
1. \( 26, 5, -16 \) or
2. \( 2, 5, 8 \)