The 18th term of an A.P. exceeds its 12th term by 24. Find the common difference.

Answer

To find the common difference of the arithmetic progression (A.P.) where the 18th term exceeds the 12th term by 24, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the terms 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 the expressions for the 18th and 12th terms**:
- The 18th term (\( A_{18} \)):
\[ A_{18} = A + (18-1)D = A + 17D \]
- The 12th term (\( A_{12} \)):
\[ A_{12} = A + (12-1)D = A + 11D \]

3. **Set up the equation based on the problem statement**:
According to the problem, the 18th term exceeds the 12th term by 24:
\[ A_{18} = A_{12} + 24 \]
Substituting the expressions for \( A_{18} \) and \( A_{12} \):
\[ A + 17D = (A + 11D) + 24 \]

4. **Simplify the equation**:
\[ A + 17D = A + 11D + 24 \]
Now, subtract \( A \) from both sides:
\[ 17D = 11D + 24 \]

5. **Isolate the common difference \( D \)**:
Subtract \( 11D \) from both sides:
\[ 17D - 11D = 24 \]
This simplifies to:
\[ 6D = 24 \]

6. **Solve for \( D \)**:
Divide both sides by 6:
\[ D = \frac{24}{6} = 4 \]

### Conclusion:
The common difference \( D \) of the A.P. is **4**.

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