If numbers n- 2, 4n -1 and 5n + 2 are in A.P. find the value of n and its next two terms.

Answer

To solve the problem, we need to find the value of \( n \) such that the numbers \( n - 2 \), \( 4n - 1 \), and \( 5n + 2 \) are in Arithmetic Progression (A.P.).

### Step-by-Step Solution:

1. **Understanding A.P.**:
For three numbers \( a \), \( b \), and \( c \) to be in A.P., the condition is:
\[ 2b = a + c \]
Here, let:
- \( a = n - 2 \)
- \( b = 4n - 1 \)
- \( c = 5n + 2 \)

2. **Setting up the equation**:
Using the A.P. condition:
\[ 2(4n - 1) = (n - 2) + (5n + 2) \]

3. **Expanding both sides**:
- Left-hand side:
\[ 2(4n - 1) = 8n - 2 \]
- Right-hand side:
\[ (n - 2) + (5n + 2) = n - 2 + 5n + 2 = 6n \]

4. **Equating both sides**:
Now we have:
\[ 8n - 2 = 6n \]

5. **Solving for \( n \)**:
Rearranging the equation:
\[ 8n - 6n = 2 \]
\[ 2n = 2 \]
\[ n = 1 \]

6. **Finding the terms**:
Now that we have \( n = 1 \), we can find the three terms:
- First term:
\[ n - 2 = 1 - 2 = -1 \]
- Second term:
\[ 4n - 1 = 4(1) - 1 = 3 \]
- Third term:
\[ 5n + 2 = 5(1) + 2 = 7 \]

So the three terms are \( -1, 3, 7 \).

7. **Finding the common difference**:
The common difference \( d \) is given by:
\[ d = \text{second term} - \text{first term} = 3 - (-1) = 3 + 1 = 4 \]

8. **Finding the next two terms**:
- Next term after \( 7 \):
\[ 7 + 4 = 11 \]
- Next term after \( 11 \):
\[ 11 + 4 = 15 \]

### Final Answer:
The value of \( n \) is \( 1 \), and the next two terms after \( -1, 3, 7 \) are \( 11 \) and \( 15 \).