if 9 times the 9 th term of an AP is equal to 13 times the 13 th term , then the 22 nd term of the AP is

Options

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

Explanation

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

### Step-by-step Solution:

1. **Understanding the General 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 and \( d \) is the common difference.

2. **Expressing the 9th and 13th Terms**:
- The 9th term (\( a_9 \)) can be written as:
\[ a_9 = a + (9-1)d = a + 8d \]
- The 13th term (\( a_{13} \)) can be written as:
\[ a_{13} = a + (13-1)d = a + 12d \]

3. **Setting Up the Equation**:
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. **Expanding Both Sides**:
Expanding the left and right sides gives:
\[ 9a + 72d = 13a + 156d \]

5. **Rearranging the Equation**:
Rearranging the equation to isolate \( a \) and \( d \):
\[ 9a - 13a = 156d - 72d \]
This simplifies to:
\[ -4a = 84d \]

6. **Solving for \( a \)**:
Dividing both sides by -4:
\[ a = -\frac{84}{4}d = -21d \]

7. **Finding the 22nd Term**:
The 22nd term (\( a_{22} \)) can be expressed as:
\[ a_{22} = a + (22-1)d = a + 21d \]
Substituting the value of \( a \):
\[ a_{22} = -21d + 21d = 0 \]

### Final Answer:
The 22nd term of the AP is:
\[ \boxed{0} \]