The sum of the first three terms of an Arithmeic Progression (A.P.) is 42 and the product of the first and third term is 52. Find the first term and the common difference.

Answer

To solve the problem, we need to find the first term and the common difference of an Arithmetic Progression (A.P.) given two conditions:

1. The sum of the first three terms is 42.
2. The product of the first and third terms is 52.

Let's denote the first term of the A.P. as \( a \) and the common difference as \( d \). Therefore, the first three terms of the A.P. can be expressed as:
- First term: \( a \)
- Second term: \( a + d \)
- Third term: \( a + 2d \)

### Step 1: Set up the equations based on the conditions

From the first condition (sum of the first three terms):
\[ a + (a + d) + (a + 2d) = 42 \]
This simplifies to:
\[ 3a + 3d = 42 \]
Dividing the entire equation by 3 gives:
\[ a + d = 14 \quad \text{(Equation 1)} \]

From the second condition (product of the first and third terms):
\[ a \cdot (a + 2d) = 52 \]
Expanding this gives:
\[ a^2 + 2ad = 52 \quad \text{(Equation 2)} \]

### Step 2: Solve for \( d \) in terms of \( a \)

From Equation 1, we can express \( d \) in terms of \( a \):
\[ d = 14 - a \]

### Step 3: Substitute \( d \) into Equation 2

Substituting \( d = 14 - a \) into Equation 2:
\[ a^2 + 2a(14 - a) = 52 \]
Expanding this gives:
\[ a^2 + 28a - 2a^2 = 52 \]
Combining like terms results in:
\[ -a^2 + 28a - 52 = 0 \]
Multiplying through by -1 gives:
\[ a^2 - 28a + 52 = 0 \]

### Step 4: Solve the quadratic equation

Now we can use the quadratic formula to solve for \( a \):
\[ a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -28 \), and \( c = 52 \):
\[ a = \frac{28 \pm \sqrt{(-28)^2 - 4 \cdot 1 \cdot 52}}{2 \cdot 1} \]
Calculating the discriminant:
\[ = \frac{28 \pm \sqrt{784 - 208}}{2} \]
\[ = \frac{28 \pm \sqrt{576}}{2} \]
\[ = \frac{28 \pm 24}{2} \]
This gives us two possible values for \( a \):
1. \( a = \frac{52}{2} = 26 \)
2. \( a = \frac{4}{2} = 2 \)

### Step 5: Find corresponding \( d \) values

Now we substitute back to find \( d \) for each value of \( a \):

1. If \( a = 26 \):
\[ d = 14 - 26 = -12 \]

2. If \( a = 2 \):
\[ d = 14 - 2 = 12 \]

### Conclusion

The two sets of solutions are:
1. First term \( a = 26 \), common difference \( d = -12 \)
2. First term \( a = 2 \), common difference \( d = 12 \)

Since the problem asks for the first term and the common difference, we can conclude:
- First term: \( 2 \)
- Common difference: \( 12 \)