The fourth term of an A.P. is equal to 3 times the first term, and the seventh term exceeds twice the third term by 1. Find the first term and the common difference.

Answer

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

### Step 1: Understand the given information
We know:
1. The fourth term of the A.P. is equal to 3 times the first term.
2. The seventh term exceeds twice the third term by 1.

Let:
- \( a \) = first term of the A.P.
- \( d \) = common difference of the A.P.

### Step 2: Write the formulas for the terms
The \( n \)-th term of an A.P. is given by:
\[ a_n = a + (n-1)d \]

Using this formula:
- The fourth term \( a_4 \) is:
\[ a_4 = a + (4-1)d = a + 3d \]
- The seventh term \( a_7 \) is:
\[ a_7 = a + (7-1)d = a + 6d \]
- The third term \( a_3 \) is:
\[ a_3 = a + (3-1)d = a + 2d \]

### Step 3: Set up the equations based on the given information
From the first piece of information:
\[ a + 3d = 3a \quad \text{(1)} \]

From the second piece of information:
\[ a + 6d = 2(a + 2d) + 1 \quad \text{(2)} \]

### Step 4: Simplify the equations
**From Equation (1)**:
\[ a + 3d = 3a \]
Rearranging gives:
\[ 3d = 3a - a \implies 3d = 2a \implies a = \frac{3d}{2} \quad \text{(3)} \]

**From Equation (2)**:
\[ a + 6d = 2(a + 2d) + 1 \]
Expanding the right side:
\[ a + 6d = 2a + 4d + 1 \]
Rearranging gives:
\[ a + 6d - 4d = 2a + 1 \implies a + 2d = 2a + 1 \]
Rearranging further gives:
\[ 2d = 2a + 1 - a \implies 2d = a + 1 \quad \text{(4)} \]

### Step 5: Substitute Equation (3) into Equation (4)
Substituting \( a = \frac{3d}{2} \) into \( 2d = a + 1 \):
\[ 2d = \frac{3d}{2} + 1 \]
Multiplying through by 2 to eliminate the fraction:
\[ 4d = 3d + 2 \]
Rearranging gives:
\[ 4d - 3d = 2 \implies d = 2 \]

### Step 6: Find the first term using the value of \( d \)
Substituting \( d = 2 \) back into Equation (3):
\[ a = \frac{3d}{2} = \frac{3 \times 2}{2} = 3 \]

### Conclusion
Thus, the first term \( a \) is 3 and the common difference \( d \) is 2.

### Final Answer
- First term \( a = 3 \)
- Common difference \( d = 2 \)