For an A.P., `t_( 1) = 25` and `t_(20) = 405` . Find the common difference.

Answer

To find the common difference of the arithmetic progression (A.P.) where the first term \( t_1 = 25 \) and the 20th term \( t_{20} = 405 \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the formula for the nth term of an A.P.**:
The nth term of an arithmetic progression is given by the formula:
\[ t_n = a + (n - 1) \cdot d \]
where \( a \) is the first term, \( d \) is the common difference, and \( n \) is the term number.

2. **Substitute the known values for the 20th term**:
We know that \( t_{20} = 405 \) and \( t_1 = 25 \). Thus, substituting into the formula for the 20th term:
\[ t_{20} = a + (20 - 1) \cdot d \]
becomes:
\[ 405 = 25 + 19d \]

3. **Rearrange the equation to solve for \( d \)**:
First, subtract 25 from both sides:
\[ 405 - 25 = 19d \]
This simplifies to:
\[ 380 = 19d \]

4. **Divide both sides by 19 to find \( d \)**:
\[ d = \frac{380}{19} \]
Performing the division:
\[ d = 20 \]

5. **Conclusion**:
The common difference \( d \) of the arithmetic progression is:
\[ \boxed{20} \]