If (k - 3), (2k + 1) and (4k + 3) are three consecutive terms of an A.P., find the value of k.

Answer

To find the value of \( k \) such that \( (k - 3), (2k + 1), (4k + 3) \) are three consecutive terms of an arithmetic progression (A.P.), we can follow these steps:

### Step 1: Identify the terms
Let:
- First term \( a = k - 3 \)
- Second term \( a_2 = 2k + 1 \)
- Third term \( a_3 = 4k + 3 \)

### Step 2: Use the property of A.P.
In an arithmetic progression, the difference between consecutive terms is constant. Therefore, we can set up the following equations based on the common difference \( d \):

1. \( d = a_2 - a = (2k + 1) - (k - 3) \)
2. \( d = a_3 - a_2 = (4k + 3) - (2k + 1) \)

### Step 3: Calculate the first difference
Calculate \( d \) from the first equation:
\[ d = (2k + 1) - (k - 3) \]
Expanding this gives:
\[ d = 2k + 1 - k + 3 = k + 4 \]

### Step 4: Calculate the second difference
Now calculate \( d \) from the second equation:
\[ d = (4k + 3) - (2k + 1) \]
Expanding this gives:
\[ d = 4k + 3 - 2k - 1 = 2k + 2 \]

### Step 5: Set the two expressions for \( d \) equal
Since both expressions represent the same common difference, we can set them equal to each other:
\[ k + 4 = 2k + 2 \]

### Step 6: Solve for \( k \)
Rearranging the equation:
\[ k + 4 - 2 = 2k \]
\[ k + 2 = 2k \]
Subtract \( k \) from both sides:
\[ 2 = 2k - k \]
\[ 2 = k \]

### Conclusion
Thus, the value of \( k \) is:
\[ \boxed{2} \]