If the lines `a x+2y+1=0,b x+3y+1=0a n dc x+4y+1=0` are concurrent, then `a ,b ,c` are a. A.P. b. G.P. c. H.P. d. none of these

Explanation

To determine the relationship between \( a, b, c \) when the lines \( ax + 2y + 1 = 0 \), \( bx + 3y + 1 = 0 \), and \( cx + 4y + 1 = 0 \) are concurrent, we can follow these steps:

### Step-by-Step Solution:

1. **Set Up the Equations**:
We have three equations of lines:
\[ ax + 2y + 1 = 0 \quad (1) \]
\[ bx + 3y + 1 = 0 \quad (2) \]
\[ cx + 4y + 1 = 0 \quad (3) \]

2. **Express y in terms of x**:
Rearranging each equation to express \( y \) in terms of \( x \):
- From equation (1):
\[ 2y = -ax - 1 \implies y = -\frac{a}{2}x - \frac{1}{2} \quad (4) \]
- From equation (2):
\[ 3y = -bx - 1 \implies y = -\frac{b}{3}x - \frac{1}{3} \quad (5) \]
- From equation (3):
\[ 4y = -cx - 1 \implies y = -\frac{c}{4}x - \frac{1}{4} \quad (6) \]

3. **Find the Concurrent Point**:
For the lines to be concurrent, they must intersect at a common point \( (p, q) \). Substituting \( p \) and \( q \) into the equations:
- For equation (1):
\[ ap + 2q + 1 = 0 \implies ap + 2q = -1 \quad (7) \]
- For equation (2):
\[ bp + 3q + 1 = 0 \implies bp + 3q = -1 \quad (8) \]
- For equation (3):
\[ cp + 4q + 1 = 0 \implies cp + 4q = -1 \quad (9) \]

4. **Subtract Equations**:
Now, we will subtract equation (8) from equation (7):
\[ (ap + 2q) - (bp + 3q) = 0 \implies (a - b)p - q = 0 \quad (10) \]
This implies:
\[ (a - b)p = q \quad (11) \]

Next, subtract equation (9) from equation (8):
\[ (bp + 3q) - (cp + 4q) = 0 \implies (b - c)p - q = 0 \quad (12) \]
This implies:
\[ (b - c)p = q \quad (13) \]

5. **Equate the Two Expressions for q**:
From equations (11) and (13), we have:
\[ (a - b)p = (b - c)p \]
Since \( p \neq 0 \) (assuming the lines are not parallel), we can divide both sides by \( p \):
\[ a - b = b - c \quad (14) \]

6. **Rearranging the Equation**:
Rearranging equation (14):
\[ a + c = 2b \quad (15) \]
This shows that \( a, b, c \) are in Arithmetic Progression (A.P.).

### Conclusion:
Thus, the values of \( a, b, c \) are in A.P., which corresponds to option (a).