Three numbers a, b and c are in between 2 and 18 such that 2, a, b are in A.P. and b, c, 18 are in G.P . If `a+b+c=25`, then the value of `c-a` is

Options

A. 4
B. 3
C. 7
D. 0

Answer

To solve the problem, we need to find the values of \( a \), \( b \), and \( c \) given the conditions of the problem. Let's break it down step by step.

### Step 1: Understand the conditions
We know that:
1. \( 2, a, b \) are in Arithmetic Progression (A.P.)
2. \( b, c, 18 \) are in Geometric Progression (G.P.)
3. \( a + b + c = 25 \)

### Step 2: Express \( a \) in terms of \( b \)
Since \( 2, a, b \) are in A.P., we can write:
\[ a = \frac{2 + b}{2} \]

### Step 3: Express \( c \) in terms of \( b \)
Since \( b, c, 18 \) are in G.P., we can write:
\[ c = \sqrt{b \cdot 18} = 3\sqrt{2b} \]

### Step 4: Substitute \( a \) and \( c \) into the equation \( a + b + c = 25 \)
Substituting the expressions for \( a \) and \( c \):
\[ \frac{2 + b}{2} + b + 3\sqrt{2b} = 25 \]

### Step 5: Clear the fraction by multiplying through by 2
Multiplying the entire equation by 2 to eliminate the fraction:
\[ 2 + b + 2b + 6\sqrt{2b} = 50 \]
This simplifies to:
\[ 3b + 6\sqrt{2b} = 48 \]

### Step 6: Rearrange the equation
Rearranging gives:
\[ 3b + 6\sqrt{2b} - 48 = 0 \]

### Step 7: Isolate the square root term
We can isolate the square root term:
\[ 6\sqrt{2b} = 48 - 3b \]

### Step 8: Square both sides to eliminate the square root
Squaring both sides:
\[ 36 \cdot 2b = (48 - 3b)^2 \]
This expands to:
\[ 72b = 2304 - 288b + 9b^2 \]

### Step 9: Rearrange into a standard quadratic equation
Rearranging gives:
\[ 9b^2 - 360b + 2304 = 0 \]

### Step 10: Solve the quadratic equation using the quadratic formula
Using the quadratic formula \( b = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \):
Here, \( A = 9 \), \( B = -360 \), and \( C = 2304 \).
Calculating the discriminant:
\[ B^2 - 4AC = (-360)^2 - 4 \cdot 9 \cdot 2304 = 129600 - 82944 = 46656 \]
Now, calculating \( b \):
\[ b = \frac{360 \pm \sqrt{46656}}{18} \]
Calculating \( \sqrt{46656} = 216 \):
\[ b = \frac{360 \pm 216}{18} \]
This gives us two possible values for \( b \):
1. \( b = \frac{576}{18} = 32 \) (not valid since \( b \) must be between 2 and 18)
2. \( b = \frac{144}{18} = 8 \)

### Step 11: Find \( a \) and \( c \)
Now substituting \( b = 8 \) back into the equations for \( a \) and \( c \):
\[ a = \frac{2 + 8}{2} = 5 \]
\[ c = 3\sqrt{2 \cdot 8} = 3\sqrt{16} = 12 \]

### Step 12: Calculate \( c - a \)
Finally, we find:
\[ c - a = 12 - 5 = 7 \]

### Final Answer
The value of \( c - a \) is \( \boxed{7} \).