If p, 2p - 1, 2p + 1 are three consecutive terms of an A.P., then what is the value of p ?

Answer

To find the value of \( p \) given that \( p, 2p - 1, 2p + 1 \) are three consecutive terms of an arithmetic progression (A.P.), we can use the property of A.P. that states the double of the middle term is equal to the sum of the first and third terms.

### Step-by-Step Solution:

1. **Identify the terms**: The three terms are:
- First term: \( a = p \)
- Second term: \( b = 2p - 1 \)
- Third term: \( c = 2p + 1 \)

2. **Apply the A.P. property**: According to the property of A.P., we have:
\[ 2b = a + c \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ 2(2p - 1) = p + (2p + 1) \]

3. **Simplify the left side**: Calculate \( 2(2p - 1) \):
\[ 4p - 2 \]

4. **Simplify the right side**: Calculate \( p + (2p + 1) \):
\[ p + 2p + 1 = 3p + 1 \]

5. **Set the two sides equal**: Now we have the equation:
\[ 4p - 2 = 3p + 1 \]

6. **Rearrange the equation**: To isolate \( p \), subtract \( 3p \) from both sides:
\[ 4p - 3p - 2 = 1 \]
This simplifies to:
\[ p - 2 = 1 \]

7. **Solve for \( p \)**: Add 2 to both sides:
\[ p = 1 + 2 \]
Thus, we find:
\[ p = 3 \]

### Final Answer:
The value of \( p \) is \( 3 \).