The `7^(th)` term of an A.P. exceeds its `10^(th)` term by 7. Find the common difference.

Answer

To solve the problem step by step, we will follow the given information about the arithmetic progression (A.P.) and apply the general formula for the nth term of an A.P.

### Step-by-Step Solution:

1. **Understand the Problem Statement:**
We are given that the 7th term of an A.P. exceeds its 10th term by 7. We need to find the common difference (d) of the A.P.

2. **Write the General Formula for the nth Term:**
The nth term of an A.P. can be expressed as:
\[ A_n = A + (n - 1) \cdot d \]
where \( A \) is the first term, \( n \) is the term number, and \( d \) is the common difference.

3. **Express the 7th and 10th Terms:**
Using the formula, we can write:
- The 7th term (\( A_7 \)):
\[ A_7 = A + (7 - 1) \cdot d = A + 6d \]
- The 10th term (\( A_{10} \)):
\[ A_{10} = A + (10 - 1) \cdot d = A + 9d \]

4. **Set Up the Equation Based on the Given Information:**
According to the problem, the 7th term exceeds the 10th term by 7:
\[ A_7 = A_{10} + 7 \]
Substituting the expressions for \( A_7 \) and \( A_{10} \):
\[ A + 6d = (A + 9d) + 7 \]

5. **Simplify the Equation:**
Rearranging the equation:
\[ A + 6d = A + 9d + 7 \]
We can cancel \( A \) from both sides:
\[ 6d = 9d + 7 \]

6. **Isolate the Common Difference (d):**
Rearranging gives:
\[ 6d - 9d = 7 \]
\[ -3d = 7 \]
Dividing both sides by -3:
\[ d = -\frac{7}{3} \]

7. **Conclusion:**
The common difference \( d \) of the A.P. is:
\[ d = -\frac{7}{3} \]