If 7 times the seventh term of the AP is equal to 5 times the fifth term, then find the value of its 12th term.

Options

A. `4`
B. `2`
C. `0`
D. None

Explanation

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

### Step-by-Step Solution:

1. **Understanding the Problem**: We are given that 7 times the 7th term of an AP is equal to 5 times the 5th term. We need to find the 12th term of this AP.

2. **Using the Formula for the nth Term of an AP**: The nth term of an AP can be expressed as:
\[ A_n = A + (n - 1)D \]
where \( A \) is the first term, \( D \) is the common difference, and \( n \) is the term number.

3. **Expressing the 7th and 5th Terms**:
- The 7th term \( A_7 \) can be expressed as:
\[ A_7 = A + (7 - 1)D = A + 6D \]
- The 5th term \( A_5 \) can be expressed as:
\[ A_5 = A + (5 - 1)D = A + 4D \]

4. **Setting Up the Equation**: According to the problem statement:
\[ 7 \times A_7 = 5 \times A_5 \]
Substituting the expressions for \( A_7 \) and \( A_5 \):
\[ 7(A + 6D) = 5(A + 4D) \]

5. **Expanding Both Sides**:
\[ 7A + 42D = 5A + 20D \]

6. **Rearranging the Equation**: Move all terms involving \( A \) and \( D \) to one side:
\[ 7A - 5A + 42D - 20D = 0 \]
This simplifies to:
\[ 2A + 22D = 0 \]

7. **Factoring Out Common Terms**: We can factor out 2:
\[ 2(A + 11D) = 0 \]
Since \( 2 \neq 0 \), we can conclude:
\[ A + 11D = 0 \]
Thus:
\[ A = -11D \]

8. **Finding the 12th Term**: Now we can find the 12th term \( A_{12} \):
\[ A_{12} = A + (12 - 1)D = A + 11D \]
Substituting \( A = -11D \):
\[ A_{12} = -11D + 11D = 0 \]

### Final Answer:
The value of the 12th term of the AP is:
\[ \boxed{0} \]