If the numbers `3k+4,7k+1and12k-5` are in A.P., then the value of k is

Options

A. 2
B. 3
C. 4
D. 5

Answer

To solve the problem, we need to determine the value of \( k \) such that the numbers \( 3k + 4 \), \( 7k + 1 \), and \( 12k - 5 \) are in arithmetic progression (A.P.).

### Step-by-step Solution:

1. **Understanding A.P. Condition**:
For three numbers \( a_1, a_2, a_3 \) to be in A.P., the condition is:
\[ a_2 - a_1 = a_3 - a_2 \]
Here, \( a_1 = 3k + 4 \), \( a_2 = 7k + 1 \), and \( a_3 = 12k - 5 \).

2. **Setting Up the Equation**:
Using the A.P. condition:
\[ (7k + 1) - (3k + 4) = (12k - 5) - (7k + 1) \]

3. **Simplifying the Left Side**:
Simplifying the left side:
\[ 7k + 1 - 3k - 4 = 4k - 3 \]

4. **Simplifying the Right Side**:
Simplifying the right side:
\[ 12k - 5 - 7k - 1 = 5k - 6 \]

5. **Setting the Two Sides Equal**:
Now we have:
\[ 4k - 3 = 5k - 6 \]

6. **Rearranging the Equation**:
Rearranging gives:
\[ 4k - 5k = -6 + 3 \]
This simplifies to:
\[ -k = -3 \]

7. **Solving for k**:
Multiplying both sides by -1 gives:
\[ k = 3 \]

### Conclusion:
The value of \( k \) is \( 3 \).