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 find the value of \( k \) such that \( k + 1, 2k + 1, \) and \( k + 7 \) are in an Arithmetic Progression (A.P.), we follow these steps:

### Step 1: Set up the condition for A.P.
In an A.P., the difference between consecutive terms is constant. Therefore, we can set up the equation:
\[ (2k + 1) - (k + 1) = (k + 7) - (2k + 1) \]

### Step 2: Simplify the left-hand side
Calculating the left-hand side:
\[ (2k + 1) - (k + 1) = 2k + 1 - k - 1 = k \]

### Step 3: Simplify the right-hand side
Now, calculating the right-hand side:
\[ (k + 7) - (2k + 1) = k + 7 - 2k - 1 = -k + 6 \]

### Step 4: Set the two sides equal
Now we set the two sides equal to each other:
\[ k = -k + 6 \]

### Step 5: Solve for \( k \)
To solve for \( k \), we add \( k \) to both sides:
\[ k + k = 6 \implies 2k = 6 \implies k = 3 \]

### Step 6: Substitute \( k \) back to find the terms
Now that we have \( k = 3 \), we substitute it back into the expressions for the terms:
- First term: \( k + 1 = 3 + 1 = 4 \)
- Second term: \( 2k + 1 = 2(3) + 1 = 6 + 1 = 7 \)
- Third term: \( k + 7 = 3 + 7 = 10 \)

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

### Step 7: Find the common difference
The common difference \( d \) can be calculated as:
\[ d = 7 - 4 = 3 \]

### Step 8: Find the next two terms
To find the next two terms of the A.P., we add the common difference to the last term:
- Fourth term: \( 10 + 3 = 13 \)
- Fifth term: \( 13 + 3 = 16 \)

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

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