What is the common difference of an A.P. in which `a_(21) - a_(7) = 82`?

Answer

To find the common difference \( d \) of the arithmetic progression (A.P.) given that \( a_{21} - a_{7} = 82 \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the formula for the nth term of an A.P.**:
The nth term of an A.P. can be expressed as:
\[ a_n = a + (n-1)d \]
where \( a \) is the first term, \( d \) is the common difference, and \( n \) is the term number.

2. **Write expressions for \( a_{21} \) and \( a_{7} \)**:
Using the formula for the nth term:
- For \( a_{21} \):
\[ a_{21} = a + (21-1)d = a + 20d \]
- For \( a_{7} \):
\[ a_{7} = a + (7-1)d = a + 6d \]

3. **Set up the equation based on the given information**:
We know that:
\[ a_{21} - a_{7} = 82 \]
Substituting the expressions we found:
\[ (a + 20d) - (a + 6d) = 82 \]

4. **Simplify the equation**:
The \( a \) terms cancel out:
\[ 20d - 6d = 82 \]
This simplifies to:
\[ 14d = 82 \]

5. **Solve for \( d \)**:
Divide both sides by 14:
\[ d = \frac{82}{14} \]
Simplifying this fraction:
\[ d = \frac{41}{7} \]

### Final Answer:
The common difference \( d \) of the A.P. is:
\[ \boxed{\frac{41}{7}} \]