If the first term of an A.P. is `a` and `n t h` term is `b` , then its common difference is

Options

A. `(b-a)/(n+1)`
B. `(b-a)/(n-1)`
C. `(b-a)/n`
D. `(b+a)/(n-1)`

Answer

To find the common difference of an Arithmetic Progression (A.P.) given the first term \( a \) and the \( n \)th term \( b \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the given values**:
- First term \( a \)
- \( n \)th term \( b \)

2. **Recall the formula for the \( n \)th term of an A.P.**:
The formula for the \( n \)th term \( A_n \) of an A.P. is given by:
\[ A_n = a + (n - 1) \cdot d \]
where \( d \) is the common difference.

3. **Substitute the known values into the formula**:
Since we know that the \( n \)th term \( A_n \) is \( b \), we can substitute \( b \) for \( A_n \):
\[ b = a + (n - 1) \cdot d \]

4. **Rearrange the equation to solve for \( d \)**:
To isolate \( d \), we can rearrange the equation:
\[ b - a = (n - 1) \cdot d \]

5. **Divide both sides by \( (n - 1) \)**:
To find \( d \), divide both sides of the equation by \( (n - 1) \):
\[ d = \frac{b - a}{n - 1} \]

6. **Conclusion**:
Thus, the common difference \( d \) of the A.P. is:
\[ d = \frac{b - a}{n - 1} \]