The sum of the 4th term and the 8th term of an A.P is 24 and the sum of the 6th term and the 10th term is 44. Find the first three terms of the A.P.

Answer

To solve the problem step by step, we will use the properties of an Arithmetic Progression (A.P.).

### Step 1: Define the terms of the A.P.
Let the first term of the A.P. be \( a \) and the common difference be \( d \).

The \( n \)-th term of an A.P. can be expressed as:
\[ T_n = a + (n-1)d \]

### Step 2: Write the expressions for the 4th and 8th terms.
- The 4th term \( T_4 \):
\[ T_4 = a + (4-1)d = a + 3d \]

- The 8th term \( T_8 \):
\[ T_8 = a + (8-1)d = a + 7d \]

### Step 3: Set up the equation for the sum of the 4th and 8th terms.
According to the problem, the sum of the 4th and 8th terms is 24:
\[ T_4 + T_8 = 24 \]
Substituting the expressions we found:
\[ (a + 3d) + (a + 7d) = 24 \]
This simplifies to:
\[ 2a + 10d = 24 \quad \text{(Equation 1)} \]

### Step 4: Write the expressions for the 6th and 10th terms.
- The 6th term \( T_6 \):
\[ T_6 = a + (6-1)d = a + 5d \]

- The 10th term \( T_{10} \):
\[ T_{10} = a + (10-1)d = a + 9d \]

### Step 5: Set up the equation for the sum of the 6th and 10th terms.
According to the problem, the sum of the 6th and 10th terms is 44:
\[ T_6 + T_{10} = 44 \]
Substituting the expressions we found:
\[ (a + 5d) + (a + 9d) = 44 \]
This simplifies to:
\[ 2a + 14d = 44 \quad \text{(Equation 2)} \]

### Step 6: Solve the system of equations.
Now we have two equations:
1. \( 2a + 10d = 24 \)
2. \( 2a + 14d = 44 \)

We can subtract Equation 1 from Equation 2:
\[ (2a + 14d) - (2a + 10d) = 44 - 24 \]
This simplifies to:
\[ 4d = 20 \]
Dividing both sides by 4 gives:
\[ d = 5 \]

### Step 7: Substitute \( d \) back into one of the equations to find \( a \).
Using Equation 1:
\[ 2a + 10(5) = 24 \]
This simplifies to:
\[ 2a + 50 = 24 \]
Subtracting 50 from both sides gives:
\[ 2a = 24 - 50 \]
\[ 2a = -26 \]
Dividing both sides by 2 gives:
\[ a = -13 \]

### Step 8: Find the first three terms of the A.P.
Now that we have \( a \) and \( d \):
- First term \( T_1 = a = -13 \)
- Second term \( T_2 = a + d = -13 + 5 = -8 \)
- Third term \( T_3 = a + 2d = -13 + 2(5) = -13 + 10 = -3 \)

Thus, the first three terms of the A.P. are:
\[ -13, -8, -3 \]

### Final Answer:
The first three terms of the A.P. are \( -13, -8, -3 \).