The value of p for which (2p + 1), 10 and (5p + 5) are three consecutive terms of an AP, is :

Options

A. `-1`
B. `-2`
C. 1
D. 2

Answer

To find the value of \( p \) for which \( (2p + 1) \), \( 10 \), and \( (5p + 5) \) are three consecutive terms of an Arithmetic Progression (AP), we can follow these steps:

### Step 1: Understand the property of AP
In an AP, the difference between consecutive terms is constant. Therefore, for three terms \( a \), \( b \), and \( c \), the condition is:
\[ b - a = c - b \]

### Step 2: Assign the terms
Let:
- First term \( a = 2p + 1 \)
- Second term \( b = 10 \)
- Third term \( c = 5p + 5 \)

### Step 3: Set up the equation using the AP property
Using the property of AP, we can write:
\[ 10 - (2p + 1) = (5p + 5) - 10 \]

### Step 4: Simplify the left side
Calculate the left side:
\[ 10 - (2p + 1) = 10 - 2p - 1 = 9 - 2p \]

### Step 5: Simplify the right side
Calculate the right side:
\[ (5p + 5) - 10 = 5p + 5 - 10 = 5p - 5 \]

### Step 6: Set the two sides equal
Now we have:
\[ 9 - 2p = 5p - 5 \]

### Step 7: Rearrange the equation
Rearranging gives:
\[ 9 + 5 = 5p + 2p \]
\[ 14 = 7p \]

### Step 8: Solve for \( p \)
Now, divide both sides by 7:
\[ p = \frac{14}{7} = 2 \]

### Conclusion
Thus, the value of \( p \) for which \( (2p + 1) \), \( 10 \), and \( (5p + 5) \) are three consecutive terms of an AP is:
\[ \boxed{2} \]