Find the value of k so that 8k +4, 6k-2, and 2k + 7 will form an A.P.

Answer

To find the value of \( k \) such that the terms \( 8k + 4 \), \( 6k - 2 \), and \( 2k + 7 \) form an arithmetic progression (A.P.), we will use the property of A.P. which states that for three terms \( a \), \( b \), and \( c \) to be in A.P., the following condition must hold:

\[ 2b = a + c \]

Here, we can identify:
- \( a = 8k + 4 \)
- \( b = 6k - 2 \)
- \( c = 2k + 7 \)

Now, we will substitute these expressions into the A.P. condition:

1. **Step 1: Set up the equation using the A.P. condition.**
\[ 2(6k - 2) = (8k + 4) + (2k + 7) \]

2. **Step 2: Simplify both sides.**
- Left side:
\[ 2(6k - 2) = 12k - 4 \]
- Right side:
\[ (8k + 4) + (2k + 7) = 8k + 2k + 4 + 7 = 10k + 11 \]

So, we have:
\[ 12k - 4 = 10k + 11 \]

3. **Step 3: Rearrange the equation to isolate \( k \).**
\[ 12k - 10k = 11 + 4 \]
\[ 2k = 15 \]

4. **Step 4: Solve for \( k \).**
\[ k = \frac{15}{2} \]

Thus, the value of \( k \) is \( \frac{15}{2} \).