Which term of the A.P. 21, 42, 63, ... is 420 ?

Answer

To find which term of the arithmetic progression (A.P.) 21, 42, 63, ... is equal to 420, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the first term (a) and the common difference (d)**:
- The first term \( a = 21 \).
- The common difference \( d \) can be calculated as the difference between the second term and the first term:
\[ d = 42 - 21 = 21 \]

2. **Use the formula for the nth term of an A.P.**:
- The formula for the nth term of an A.P. is given by:
\[ a_n = a + (n - 1) \cdot d \]
- We need to find \( n \) such that \( a_n = 420 \).

3. **Set up the equation**:
- Substitute the known values into the formula:
\[ 420 = 21 + (n - 1) \cdot 21 \]

4. **Simplify the equation**:
- First, subtract 21 from both sides:
\[ 420 - 21 = (n - 1) \cdot 21 \]
\[ 399 = (n - 1) \cdot 21 \]

5. **Solve for \( n - 1 \)**:
- Divide both sides by 21:
\[ n - 1 = \frac{399}{21} \]
- Calculate \( \frac{399}{21} \):
\[ n - 1 = 19 \]

6. **Find \( n \)**:
- Add 1 to both sides:
\[ n = 19 + 1 = 20 \]

7. **Conclusion**:
- Therefore, the term 420 is the 20th term of the A.P.

### Final Answer:
The 20th term of the A.P. 21, 42, 63, ... is 420.