If 7 times the seventh term of an Arithmetic progression is equal to 11 times its eleventh term, then the 18th term of the AP will be

Options

A. 1
B. 0
C. 2
D. -1

Answer

To solve the problem, we need to find the 18th term of an arithmetic progression (AP) given that 7 times the seventh term is equal to 11 times the eleventh term.

### Step-by-Step Solution:

1. **Define the nth term of an AP**:
The nth term of an arithmetic progression can be expressed as:
\[ T_n = a + (n-1)d \]
where \( a \) is the first term and \( d \) is the common difference.

2. **Express the 7th and 11th terms**:
Using the formula for the nth term:
- The 7th term \( T_7 \):
\[ T_7 = a + (7-1)d = a + 6d \]
- The 11th term \( T_{11} \):
\[ T_{11} = a + (11-1)d = a + 10d \]

3. **Set up the equation based on the problem statement**:
According to the problem, we have:
\[ 7 \cdot T_7 = 11 \cdot T_{11} \]
Substituting the expressions for \( T_7 \) and \( T_{11} \):
\[ 7(a + 6d) = 11(a + 10d) \]

4. **Expand both sides**:
Expanding the left side:
\[ 7a + 42d = 11a + 110d \]

5. **Rearrange the equation**:
Bringing all terms involving \( a \) to one side and all terms involving \( d \) to the other side:
\[ 7a + 42d - 11a - 110d = 0 \]
Simplifying this gives:
\[ -4a - 68d = 0 \]
or
\[ 4a + 68d = 0 \]

6. **Factor out the common term**:
Dividing the entire equation by 4:
\[ a + 17d = 0 \]
This implies:
\[ a = -17d \]

7. **Find the 18th term**:
Now, we need to find the 18th term \( T_{18} \):
\[ T_{18} = a + (18-1)d = a + 17d \]
Substituting \( a = -17d \) into the equation:
\[ T_{18} = -17d + 17d = 0 \]

### Final Answer:
The 18th term of the AP is \( 0 \).