If a, b, c are in AP amd b, c, a are in GP , then show that c, a, b are in H.P. Find ` a : b : c` .

Answer

To solve the problem, we need to show that if \( a, b, c \) are in Arithmetic Progression (AP) and \( b, c, a \) are in Geometric Progression (GP), then \( c, a, b \) are in Harmonic Progression (HP). We also need to find the ratio \( a : b : c \).

### Step-by-Step Solution:

1. **Understanding the Conditions**:
- Since \( a, b, c \) are in AP, we have:
\[ 2b = a + c \quad \text{(1)} \]
- Since \( b, c, a \) are in GP, we have:
\[ c^2 = ab \quad \text{(2)} \]

2. **Expressing \( c \) in terms of \( a \) and \( b \)**:
- From equation (1), we can express \( c \):
\[ c = 2b - a \quad \text{(3)} \]

3. **Substituting \( c \) into the GP condition**:
- Substitute equation (3) into equation (2):
\[ (2b - a)^2 = ab \]
- Expanding the left side:
\[ 4b^2 - 4ab + a^2 = ab \]
- Rearranging gives:
\[ 4b^2 - 5ab + a^2 = 0 \quad \text{(4)} \]

4. **Solving the quadratic equation**:
- The equation (4) is a quadratic in terms of \( a \):
\[ a^2 - 5ab + 4b^2 = 0 \]
- Using the quadratic formula \( a = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \):
- Here, \( A = 1, B = -5b, C = 4b^2 \):
\[ a = \frac{5b \pm \sqrt{(-5b)^2 - 4 \cdot 1 \cdot 4b^2}}{2 \cdot 1} \]
\[ a = \frac{5b \pm \sqrt{25b^2 - 16b^2}}{2} \]
\[ a = \frac{5b \pm 3b}{2} \]
- This gives us two possible values for \( a \):
\[ a = \frac{8b}{2} = 4b \quad \text{or} \quad a = \frac{2b}{2} = b \]

5. **Finding corresponding values of \( c \)**:
- If \( a = 4b \):
\[ c = 2b - 4b = -2b \]
- If \( a = b \):
\[ c = 2b - b = b \]

6. **Checking if \( c, a, b \) are in HP**:
- For \( c, a, b \) to be in HP, the condition is:
\[ \frac{1}{a} = \frac{1}{b} + \frac{1}{c} \]
- **Case 1**: \( a = 4b \), \( c = -2b \):
- Check:
\[ \frac{1}{4b} \neq \frac{1}{b} + \frac{1}{-2b} \quad \text{(not in HP)} \]
- **Case 2**: \( a = b \), \( c = b \):
- Check:
\[ \frac{1}{b} = \frac{1}{b} + \frac{1}{b} \quad \text{(in HP)} \]

7. **Conclusion**:
- The only valid case is when \( a = b = c \).
- Thus, the ratio \( a : b : c = 1 : 1 : 1 \).