Which term of an AP : 21, 42, 63, 84, … is 210 ?

Options

A. 9th
B. 10th
C. 11th
D. 12th

Explanation

To find which term of the arithmetic progression (AP) 21, 42, 63, 84, ... is equal to 210, we will use the formula for the nth term of an AP:

### Step-by-Step Solution:

1. **Identify the first term (a) and the common difference (d)**:
- The first term \( a = 21 \).
- To find the common difference \( d \), we subtract the first term from the second term:
\[ d = 42 - 21 = 21. \]

2. **Use the formula for the nth term of an AP**:
- The formula for the nth term \( a_n \) of an AP is given by:
\[ a_n = a + (n - 1) \cdot d. \]
- We are given that \( a_n = 210 \). So we can set up the equation:
\[ 210 = 21 + (n - 1) \cdot 21. \]

3. **Rearranging the equation**:
- Subtract \( 21 \) from both sides:
\[ 210 - 21 = (n - 1) \cdot 21. \]
- This simplifies to:
\[ 189 = (n - 1) \cdot 21. \]

4. **Solve for \( n - 1 \)**:
- Divide both sides by \( 21 \):
\[ n - 1 = \frac{189}{21}. \]
- Simplifying \( \frac{189}{21} \):
\[ n - 1 = 9. \]

5. **Find the value of \( n \)**:
- Add \( 1 \) to both sides:
\[ n = 9 + 1 = 10. \]

6. **Conclusion**:
- Therefore, the term of the AP that is equal to 210 is the **10th term**.