If 9 times the 9th term of an A.P. is equal to 13 times the 13 term, then the 22nd term of the A.P. is
(i) 0
(ii) 22
(iii) 220
(iv) 198

Options

A. 0
B. 22
C. 220
D. 198

Answer

To solve the problem, we need to find the 22nd term of an arithmetic progression (A.P.) given that 9 times the 9th term is equal to 13 times the 13th term.

### Step-by-Step Solution:

1. **Define the terms of the A.P.**:
Let \( A \) be the first term and \( D \) be the common difference of the A.P. The formula for the \( n \)-th term of an A.P. is given by:
\[ A_n = A + (n - 1)D \]

2. **Write expressions for the 9th and 13th terms**:
- The 9th term \( A_9 \) is:
\[ A_9 = A + 8D \]
- The 13th term \( A_{13} \) is:
\[ A_{13} = A + 12D \]

3. **Set up the equation based on the given condition**:
According to the problem, we have:
\[ 9 \cdot A_9 = 13 \cdot A_{13} \]
Substituting the expressions for \( A_9 \) and \( A_{13} \):
\[ 9(A + 8D) = 13(A + 12D) \]

4. **Expand both sides**:
Expanding the left side:
\[ 9A + 72D \]
Expanding the right side:
\[ 13A + 156D \]

5. **Set the equation**:
Now we can set the two sides equal to each other:
\[ 9A + 72D = 13A + 156D \]

6. **Rearrange the equation**:
Rearranging gives:
\[ 9A - 13A + 72D - 156D = 0 \]
Simplifying this leads to:
\[ -4A - 84D = 0 \]

7. **Factor out common terms**:
Dividing the entire equation by -4 gives:
\[ A + 21D = 0 \]

8. **Interpret the result**:
The expression \( A + 21D = 0 \) indicates that the 22nd term \( A_{22} \) can be expressed as:
\[ A_{22} = A + 21D \]
Since \( A + 21D = 0 \), we have:
\[ A_{22} = 0 \]

### Conclusion:
Thus, the 22nd term of the A.P. is \( 0 \).

### Final Answer:
The answer is (i) 0.