If the 4th term of an arithmetic progression is 14 and the 12th term is 70, then the first term is

Options

A. -10
B. -7
C. `+7`
D. `+10`

Answer

To find the first term of the arithmetic progression (AP), we can use the information given about the 4th and 12th terms.

### Step-by-Step Solution:

1. **Understanding the Terms of an AP**:
The n-th term of an arithmetic progression can be expressed as:
\[ A_n = A + (n-1)d \]
where \( A \) is the first term and \( d \) is the common difference.

2. **Setting Up the Equations**:
From the problem, we know:
- The 4th term \( A_4 = 14 \):
\[ A + 3d = 14 \quad \text{(1)} \]
- The 12th term \( A_{12} = 70 \):
\[ A + 11d = 70 \quad \text{(2)} \]

3. **Subtracting the Equations**:
We can eliminate \( A \) by subtracting equation (1) from equation (2):
\[ (A + 11d) - (A + 3d) = 70 - 14 \]
Simplifying this gives:
\[ 8d = 56 \]

4. **Solving for \( d \)**:
Now, divide both sides by 8:
\[ d = \frac{56}{8} = 7 \]

5. **Substituting \( d \) Back to Find \( A \)**:
Now that we have \( d \), we can substitute it back into equation (1) to find \( A \):
\[ A + 3(7) = 14 \]
This simplifies to:
\[ A + 21 = 14 \]
Subtracting 21 from both sides gives:
\[ A = 14 - 21 = -7 \]

6. **Conclusion**:
The first term \( A \) of the arithmetic progression is:
\[ \boxed{-7} \]