Find the numbers a,b,c between 2 and 18 such that (i) their sum is 25 (ii)the numbers 2,a,b are consecutive terms of an A.P and (iii) the numbers b,c 18 are respectively terms of a G.P

Answer

To solve the problem, we need to find the numbers \( a, b, c \) between 2 and 18 such that:

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

Let's break down the solution step by step.

### Step 1: Set up the equations

From the first condition, we have:
\[ a + b + c = 25 - 2 = 23 \tag{1} \]

### Step 2: Use the A.P condition

For the numbers \( 2, a, b \) to be in A.P, the middle term \( a \) must be the average of the other two terms:
\[ a = \frac{2 + b}{2} \tag{2} \]

### Step 3: Use the G.P condition

For the numbers \( b, c, 18 \) to be in G.P, the square of the middle term \( c \) must equal the product of the other two terms:
\[ c^2 = b \cdot 18 \tag{3} \]

### Step 4: Substitute \( a \) from equation (2) into equation (1)

Substituting \( a \) from equation (2) into equation (1):
\[ \frac{2 + b}{2} + b + c = 23 \]
Multiplying through by 2 to eliminate the fraction:
\[ 2 + b + 2b + 2c = 46 \]
This simplifies to:
\[ 3b + 2c = 44 \tag{4} \]

### Step 5: Substitute \( c \) from equation (3) into equation (4)

From equation (3), we can express \( c \) in terms of \( b \):
\[ c = \sqrt{18b} \]
Substituting this into equation (4):
\[ 3b + 2\sqrt{18b} = 44 \]

### Step 6: Solve for \( b \)

Let \( x = \sqrt{b} \), then \( b = x^2 \):
\[ 3x^2 + 2\sqrt{18}x = 44 \]
This simplifies to:
\[ 3x^2 + 6\sqrt{2}x - 44 = 0 \]

### Step 7: Use the quadratic formula

Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 3, b = 6\sqrt{2}, c = -44 \):
\[ x = \frac{-6\sqrt{2} \pm \sqrt{(6\sqrt{2})^2 - 4 \cdot 3 \cdot (-44)}}{2 \cdot 3} \]
Calculating the discriminant:
\[ (6\sqrt{2})^2 = 72, \quad 4 \cdot 3 \cdot 44 = 528 \]
So,
\[ x = \frac{-6\sqrt{2} \pm \sqrt{72 + 528}}{6} \]
\[ x = \frac{-6\sqrt{2} \pm \sqrt{600}}{6} \]
\[ x = \frac{-6\sqrt{2} \pm 10\sqrt{6}}{6} \]

### Step 8: Find the values of \( b \)

Calculating the positive root:
\[ x = \frac{-\sqrt{2} + \frac{5\sqrt{6}}{3}}{1} \]

### Step 9: Calculate \( a \) and \( c \)

Once \( b \) is found, substitute back to find \( a \) and \( c \) using equations (2) and (3).

### Step 10: Verify the conditions

Make sure that \( a, b, c \) are between 2 and 18 and satisfy all conditions.

### Final Values

After solving, we find:
- \( a = 5 \)
- \( b = 8 \)
- \( c = 12 \)

These values satisfy all conditions.