To solve the problem, we need to find the values of \( a \), \( b \), and \( c \) under the given conditions. Let's break it down step by step.
### Step 1: Set up the equations based on the conditions given.
We know:
1. \( a + b + c = 25 \) (the sum of the three numbers).
2. \( 2, a, b \) are in A.P. (Arithmetic Progression).
3. \( b, c, 18 \) are in G.P. (Geometric Progression).
From the A.P. condition:
- Since \( 2, a, b \) are in A.P., we can use the property of A.P.:
\[
2a = 2 + b \implies a = \frac{b + 2}{2}
\]
From the G.P. condition:
- Since \( b, c, 18 \) are in G.P., we can use the property of G.P.:
\[
c^2 = b \cdot 18 \implies c = \sqrt{18b}
\]
### Step 2: Substitute \( a \) and \( c \) in the sum equation.
Now, substitute \( a \) and \( c \) into the sum equation:
\[
\frac{b + 2}{2} + b + \sqrt{18b} = 25
\]
### Step 3: Clear the fraction and simplify.
Multiply the entire equation by 2 to eliminate the fraction:
\[
b + 2 + 2b + 2\sqrt{18b} = 50
\]
Combine like terms:
\[
3b + 2 + 2\sqrt{18b} = 50
\]
Subtract 2 from both sides:
\[
3b + 2\sqrt{18b} = 48
\]
### Step 4: Isolate \( \sqrt{18b} \).
Rearranging gives:
\[
2\sqrt{18b} = 48 - 3b
\]
Dividing by 2:
\[
\sqrt{18b} = 24 - \frac{3b}{2}
\]
### Step 5: Square both sides to eliminate the square root.
Square both sides:
\[
18b = \left(24 - \frac{3b}{2}\right)^2
\]
Expanding the right side:
\[
18b = 576 - 72b + \frac{9b^2}{4}
\]
Multiply through by 4 to eliminate the fraction:
\[
72b = 2304 - 288b + 9b^2
\]
Rearranging gives:
\[
9b^2 - 360b + 2304 = 0
\]
### Step 6: Solve the quadratic equation.
Using the quadratic formula \( b = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \):
Here, \( A = 9 \), \( B = -360 \), \( C = 2304 \).
Calculate the discriminant:
\[
B^2 - 4AC = (-360)^2 - 4 \cdot 9 \cdot 2304
\]
\[
= 129600 - 82944 = 46656
\]
Now, calculate \( b \):
\[
b = \frac{360 \pm \sqrt{46656}}{18}
\]
Calculating \( \sqrt{46656} = 216 \):
\[
b = \frac{360 \pm 216}{18}
\]
Calculating the two possible values:
1. \( b = \frac{576}{18} = 32 \) (not valid since \( b \) must be between 2 and 18).
2. \( b = \frac{144}{18} = 8 \) (valid).
### Step 7: Find \( a \) and \( c \).
Now that we have \( b = 8 \):
- Calculate \( a \):
\[
a = \frac{b + 2}{2} = \frac{8 + 2}{2} = 5
\]
- Calculate \( c \):
\[
c = \sqrt{18b} = \sqrt{18 \cdot 8} = \sqrt{144} = 12
\]
### Step 8: Calculate \( abc \).
Now we can find \( abc \):
\[
abc = 5 \cdot 8 \cdot 12 = 480
\]
### Final Answer:
Thus, the value of \( abc \) is \( \boxed{480} \).
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