If the 3rd and the 6th terms of an A.P. are 7 and 13 respectively, find the first' term and the common difference.

Answer

To solve the problem, we need to find the first term (a) and the common difference (d) of the arithmetic progression (A.P.) given that the 3rd term is 7 and the 6th term is 13.

### Step-by-Step Solution:

1. **Identify the formulas for the terms of an A.P.**:
The nth term of an A.P. can be expressed as:
\[ a_n = a + (n-1)d \]
where \( a \) is the first term and \( d \) is the common difference.

2. **Set up equations for the given terms**:
- For the 3rd term (n=3):
\[ a_3 = a + 2d = 7 \quad \text{(Equation 1)} \]
- For the 6th term (n=6):
\[ a_6 = a + 5d = 13 \quad \text{(Equation 2)} \]

3. **Subtract Equation 1 from Equation 2**:
\[ (a + 5d) - (a + 2d) = 13 - 7 \]
Simplifying this gives:
\[ 5d - 2d = 6 \implies 3d = 6 \]

4. **Solve for the common difference \( d \)**:
\[ d = \frac{6}{3} = 2 \]

5. **Substitute \( d \) back into one of the equations to find \( a \)**:
Using Equation 1:
\[ a + 2d = 7 \]
Substitute \( d = 2 \):
\[ a + 2 \times 2 = 7 \implies a + 4 = 7 \]
Therefore:
\[ a = 7 - 4 = 3 \]

6. **Final result**:
The first term \( a \) is 3 and the common difference \( d \) is 2.

### Summary:
- First term \( a = 3 \)
- Common difference \( d = 2 \)