Find three numbers a, b,c between 2 & 18 such that; O their sum is 25 @ the numbers 2, a, b are consecutive terms of an AP & Q.3 the numbers b?c?18 are consecutive terms ofa GP

Options

A. `((a+c)(3a -c))/(4a^2 c^2)`
B. `2/(bc)-1/b^2`
C. `2/(bc)-1/b^2`
D. `((a-c)(3a+c))/(4a^2c^2)`

Explanation

To solve the problem, we need to find three numbers \( a, b, c \) that satisfy the following conditions:

1. Their sum is 25:
\[ a + b + c = 25 \]
2. The numbers \( 2, a, b \) are consecutive terms of an Arithmetic Progression (AP).
3. The numbers \( b, c, 18 \) are consecutive terms of a Geometric Progression (GP).

### Step 1: Set up the equations

From the condition that \( 2, a, b \) are in AP, we can use the property of AP:
\[ a - 2 = b - a \implies 2a = b + 2 \implies a = \frac{b + 2}{2} \]
This gives us our first equation.

From the condition that \( b, c, 18 \) are in GP, we can use the property of GP:
\[ \frac{c}{b} = \frac{18}{c} \implies c^2 = 18b \]
This gives us our second equation.

### Step 2: Substitute \( a \) into the sum equation

Now we have two equations:
1. \( a = \frac{b + 2}{2} \)
2. \( c^2 = 18b \)

We can substitute \( a \) into the sum equation:
\[ \frac{b + 2}{2} + b + c = 25 \]
Multiplying through by 2 to eliminate the fraction:
\[ b + 2 + 2b + 2c = 50 \implies 3b + 2c = 48 \implies 2c = 48 - 3b \implies c = \frac{48 - 3b}{2} \]

### Step 3: Substitute \( c \) into the GP equation

Now we can substitute \( c \) into the GP equation:
\[ \left(\frac{48 - 3b}{2}\right)^2 = 18b \]
Squaring both sides:
\[ \frac{(48 - 3b)^2}{4} = 18b \]
Multiplying through by 4:
\[ (48 - 3b)^2 = 72b \]
Expanding the left side:
\[ 2304 - 288b + 9b^2 = 72b \]
Rearranging gives us a quadratic equation:
\[ 9b^2 - 360b + 2304 = 0 \]

### Step 4: Solve the quadratic equation

To solve \( 9b^2 - 360b + 2304 = 0 \), we can use the quadratic formula:
\[ b = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \]
Where \( A = 9, B = -360, C = 2304 \):
\[ b = \frac{360 \pm \sqrt{(-360)^2 - 4 \cdot 9 \cdot 2304}}{2 \cdot 9} \]
Calculating the discriminant:
\[ b = \frac{360 \pm \sqrt{129600 - 82944}}{18} = \frac{360 \pm \sqrt{46656}}{18} = \frac{360 \pm 216}{18} \]
Calculating the two possible values for \( b \):
1. \( b = \frac{576}{18} = 32 \) (not valid since it must be between 2 and 18)
2. \( b = \frac{144}{18} = 8 \)

### Step 5: Find \( a \) and \( c \)

Now that we have \( b = 8 \):
Using the equation for \( a \):
\[ a = \frac{8 + 2}{2} = \frac{10}{2} = 5 \]
Using the equation for \( c \):
\[ c = \frac{48 - 3 \cdot 8}{2} = \frac{48 - 24}{2} = \frac{24}{2} = 12 \]

### Conclusion

The values of \( a, b, c \) are:
\[ a = 5, \quad b = 8, \quad c = 12 \]