For what value of k will `k + 9, 2k - 1` and 2k + 7 are the consecutive terms of an A.P.?

Answer

To find the value of \( k \) such that \( k + 9, 2k - 1, \) and \( 2k + 7 \) are consecutive terms of an Arithmetic Progression (A.P.), we can use the property of A.P. that states the difference between consecutive terms is constant.

### Step-by-Step Solution:

1. **Identify the terms**:
Let the three terms be:
- \( a_1 = k + 9 \)
- \( a_2 = 2k - 1 \)
- \( a_3 = 2k + 7 \)

2. **Use the property of A.P.**:
For these terms to be in A.P., the following condition must hold:
\[ a_2 - a_1 = a_3 - a_2 \]

3. **Substitute the values**:
Substitute the expressions for \( a_1, a_2, \) and \( a_3 \):
\[ (2k - 1) - (k + 9) = (2k + 7) - (2k - 1) \]

4. **Simplify the left side**:
\[ 2k - 1 - k - 9 = 2k + 7 - 2k + 1 \]
This simplifies to:
\[ k - 10 = 8 \]

5. **Solve for \( k \)**:
Now, solve the equation:
\[ k - 10 = 8 \]
Add 10 to both sides:
\[ k = 18 \]

### Final Answer:
The value of \( k \) is \( 18 \).

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