Consider a `DeltaABC` satisfying `2aSin^(2)((C)/(2))+2cSin^(2)((A)/(2))=2a+2c-3b`
The sides of the triangle are in

Options

A. A. GP
B. B. AP
C. C. HP
D. D. None

Answer

To solve the problem, we need to analyze the given equation and derive the relationship between the sides of triangle \( \Delta ABC \).

### Step-by-Step Solution:

1. **Write the Given Equation**:
We start with the equation provided:
\[ 2a \sin^2\left(\frac{C}{2}\right) + 2c \sin^2\left(\frac{A}{2}\right) = 2a + 2c - 3b \]

2. **Express \( \sin^2\left(\frac{C}{2}\right) \) and \( \sin^2\left(\frac{A}{2}\right) \)**:
Using the formula for \( \sin\left(\frac{C}{2}\right) \) and \( \sin\left(\frac{A}{2}\right) \):
\[ \sin\left(\frac{C}{2}\right) = \sqrt{\frac{s - a}{s}} \quad \text{and} \quad \sin\left(\frac{A}{2}\right) = \sqrt{\frac{s - b}{s}} \]
where \( s = \frac{a + b + c}{2} \).

3. **Substituting into the Equation**:
Substitute these expressions into the original equation:
\[ 2a \left(\frac{s - a}{s}\right) + 2c \left(\frac{s - b}{s}\right) = 2a + 2c - 3b \]

4. **Simplifying the Left Side**:
The left side becomes:
\[ \frac{2a(s - a) + 2c(s - b)}{s} = 2a + 2c - 3b \]
Expanding this gives:
\[ \frac{2as - 2a^2 + 2cs - 2bc}{s} = 2a + 2c - 3b \]

5. **Clearing the Denominator**:
Multiply through by \( s \) to eliminate the fraction:
\[ 2as - 2a^2 + 2cs - 2bc = s(2a + 2c - 3b) \]

6. **Rearranging the Equation**:
Rearranging gives:
\[ 2as + 2cs - 2bc = 2as + 2cs - 3bs \]
Simplifying leads to:
\[ 2bc = 3bs \]

7. **Dividing by \( b \)** (assuming \( b \neq 0 \)):
\[ 2c = 3s \]

8. **Substituting \( s \)**:
Recall \( s = \frac{a + b + c}{2} \):
\[ 2c = 3 \left(\frac{a + b + c}{2}\right) \]
Simplifying gives:
\[ 4c = 3(a + b + c) \]

9. **Rearranging**:
Rearranging leads to:
\[ 4c - 3c = 3a + 3b \implies c = 3a + 3b \]

10. **Conclusion**:
We find that \( 2b = a + c \), which indicates that the sides of triangle \( \Delta ABC \) are in Arithmetic Progression (AP).

### Final Answer:
The sides of the triangle are in **Arithmetic Progression (AP)**.