Find three numbers a, b, c between 2 and 18 such that:
(i) their sum is 25, and
(ii) the numbers 2, a, b are consecutive terms of an arithmetic progression, and
(iii) the numbers b, c, 18 are consecutive terms of a geometric progression.

Answer

To solve the problem, we need to find three numbers \( a, b, c \) between 2 and 18 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):
\[ 2a = b + 2 \quad \text{(Equation 1)} \]

3. The numbers \( b, c, 18 \) are consecutive terms of a geometric progression (GP):
\[ c^2 = 18b \quad \text{(Equation 2)} \]

### Step 1: Express \( a \) in terms of \( b \) and \( c \)

From the first condition, we can express \( a \) as:
\[ a = 25 - b - c \quad \text{(Equation 3)} \]

### Step 2: Substitute \( a \) in Equation 1

Substituting Equation 3 into Equation 1:
\[ 2(25 - b - c) = b + 2 \]
Expanding this gives:
\[ 50 - 2b - 2c = b + 2 \]
Rearranging the equation:
\[ 50 - 2 = b + 2b + 2c \]
\[ 48 = 3b + 2c \quad \text{(Equation 4)} \]

### Step 3: Express \( c \) in terms of \( b \)

From Equation 4, we can express \( c \) in terms of \( b \):
\[ 2c = 48 - 3b \]
\[ c = \frac{48 - 3b}{2} \quad \text{(Equation 5)} \]

### Step 4: Substitute \( c \) in Equation 2

Substituting Equation 5 into Equation 2:
\[ \left(\frac{48 - 3b}{2}\right)^2 = 18b \]
Expanding this gives:
\[ \frac{(48 - 3b)^2}{4} = 18b \]
Multiplying through by 4 to eliminate the fraction:
\[ (48 - 3b)^2 = 72b \]
Expanding the left side:
\[ 2304 - 288b + 9b^2 = 72b \]
Rearranging gives:
\[ 9b^2 - 360b + 2304 = 0 \]

### Step 5: Solve the quadratic equation

Using the quadratic formula \( b = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \):
Here, \( A = 9, B = -360, C = 2304 \):
\[ b = \frac{360 \pm \sqrt{(-360)^2 - 4 \cdot 9 \cdot 2304}}{2 \cdot 9} \]
Calculating the discriminant:
\[ b = \frac{360 \pm \sqrt{129600 - 82944}}{18} \]
\[ b = \frac{360 \pm \sqrt{46656}}{18} \]
\[ b = \frac{360 \pm 216}{18} \]
Calculating the 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 6: Find \( a \) and \( c \)

Using \( b = 8 \) in Equation 3:
\[ a = 25 - 8 - c \]
Using Equation 5 to find \( c \):
\[ c = \frac{48 - 3 \cdot 8}{2} = \frac{48 - 24}{2} = \frac{24}{2} = 12 \]
Now substituting \( c = 12 \) back to find \( a \):
\[ a = 25 - 8 - 12 = 5 \]

### Final Values
Thus, the three numbers are:
\[ a = 5, \quad b = 8, \quad c = 12 \]

### Verification
1. Sum: \( 5 + 8 + 12 = 25 \) (correct)
2. AP: \( 2, 5, 8 \) → \( 2a = b + 2 \) → \( 10 = 10 \) (correct)
3. GP: \( 8, 12, 18 \) → \( c^2 = 18b \) → \( 144 = 144 \) (correct)