Find the value of `k` if `k+1, 2k+1 and k+7` are in A.P.. Also find the next two terms of the A.P.

Answer

To solve the problem, we need to find the value of \( k \) such that the numbers \( k+1 \), \( 2k+1 \), and \( k+7 \) are in Arithmetic Progression (A.P.). We will also find the next two terms of the A.P. after determining \( k \).

### Step-by-Step Solution:

1. **Understanding A.P.**:
In an arithmetic progression, the middle term is the average of the other two terms. Therefore, if \( a \), \( b \), and \( c \) are in A.P., then:
\[ 2b = a + c \]

2. **Setting up the equation**:
Here, let:
- \( a = k + 1 \)
- \( b = 2k + 1 \)
- \( c = k + 7 \)

According to the A.P. condition:
\[ 2(2k + 1) = (k + 1) + (k + 7) \]

3. **Expanding both sides**:
- Left side:
\[ 2(2k + 1) = 4k + 2 \]
- Right side:
\[ (k + 1) + (k + 7) = k + 1 + k + 7 = 2k + 8 \]

4. **Setting the equation**:
Now we set the two sides equal to each other:
\[ 4k + 2 = 2k + 8 \]

5. **Solving for \( k \)**:
- Subtract \( 2k \) from both sides:
\[ 4k - 2k + 2 = 8 \]
- This simplifies to:
\[ 2k + 2 = 8 \]
- Now, subtract 2 from both sides:
\[ 2k = 6 \]
- Finally, divide by 2:
\[ k = 3 \]

6. **Finding the terms of the A.P.**:
Now that we have \( k = 3 \), we can find the terms:
- First term:
\[ k + 1 = 3 + 1 = 4 \]
- Second term:
\[ 2k + 1 = 2(3) + 1 = 6 + 1 = 7 \]
- Third term:
\[ k + 7 = 3 + 7 = 10 \]

Thus, the terms are \( 4, 7, 10 \).

7. **Finding the next two terms**:
The common difference \( d \) can be calculated as:
\[ d = 7 - 4 = 3 \]
- Fourth term:
\[ 10 + 3 = 13 \]
- Fifth term:
\[ 13 + 3 = 16 \]

### Final Answer:
- The value of \( k \) is \( 3 \).
- The next two terms of the A.P. are \( 13 \) and \( 16 \).