Let ` A_1 , A_2 …..,A_3 ` be n arithmetic means between a and b. Then the common difference of the AP is

Options

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

Answer

To find the common difference of the arithmetic progression (AP) with n arithmetic means between two numbers \( a \) and \( b \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We have two numbers \( a \) and \( b \), and we need to find \( n \) arithmetic means between them. This means we will have a sequence that starts with \( a \), followed by \( n \) means, and ends with \( b \).

2. **Identify the Terms**: The sequence will look like this:
\[ a, A_1, A_2, \ldots, A_n, b \]
Here, \( A_1, A_2, \ldots, A_n \) are the arithmetic means.

3. **Count the Total Terms**: The total number of terms in this sequence is \( n + 2 \) (including \( a \) and \( b \)).

4. **Use the Formula for the nth Term of an AP**: The nth term of an arithmetic progression can be expressed as:
\[ A_n = a + (n - 1) \cdot d \]
where \( d \) is the common difference.

5. **Identify the Last Term**: In our case, \( b \) is the \( (n + 2) \)th term. Therefore, we can write:
\[ b = a + (n + 1) \cdot d \]

6. **Rearranging the Equation**: Now, we can rearrange this equation to find \( d \):
\[ b - a = (n + 1) \cdot d \]

7. **Solve for d**: Dividing both sides by \( n + 1 \) gives:
\[ d = \frac{b - a}{n + 1} \]

8. **Conclusion**: The common difference \( d \) of the arithmetic progression is:
\[ d = \frac{b - a}{n + 1} \]

### Final Answer:
Thus, the common difference of the AP is \( \frac{b - a}{n + 1} \).