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: Set up the condition for AP
In an AP, the middle term is the average of the other two terms. Therefore, we can set up the equation:
\[ 2 \times \text{(middle term)} = \text{(first term)} + \text{(last term)} \]
Here, the first term is \( 2p + 1 \), the middle term is \( 10 \), and the last term is \( 5p + 5 \). Thus, we have:
\[ 2 \times 10 = (2p + 1) + (5p + 5) \]

### Step 2: Simplify the equation
Now, simplify the equation:
\[ 20 = (2p + 1) + (5p + 5) \]
Combine like terms:
\[ 20 = 2p + 1 + 5p + 5 \]
\[ 20 = 7p + 6 \]

### Step 3: Isolate \( p \)
Next, we need to isolate \( p \). Subtract \( 6 \) from both sides:
\[ 20 - 6 = 7p \]
\[ 14 = 7p \]

### Step 4: Solve for \( p \)
Now, divide both sides by \( 7 \):
\[ p = \frac{14}{7} \]
\[ p = 2 \]

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