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

Options

A. G .P
B. A.P
C. H . P
D. Neither in G . P . Nor in A . P nor in H . P

Explanation

To solve the problem, we need to analyze the given equation involving the sides of triangle ABC and determine the relationship between these sides. The equation given is:

\[ 2a \sin^2\left(\frac{C}{2}\right) + 2c \sin^2\left(\frac{A}{2}\right) = 2a + 2c - 3b \]

### Step 1: Substitute the half-angle sine formulas
We know that:
\[ \sin\left(\frac{C}{2}\right) = \sqrt{\frac{s-a}{s}} \quad \text{and} \quad \sin\left(\frac{A}{2}\right) = \sqrt{\frac{s-b}{s}} \]
where \( s \) is the semi-perimeter of the triangle, given by \( s = \frac{a+b+c}{2} \).

Substituting these into the equation gives:
\[ 2a \left(\frac{s-a}{s}\right) + 2c \left(\frac{s-b}{s}\right) = 2a + 2c - 3b \]

### Step 2: Simplify the equation
Multiply both sides by \( s \) to eliminate the denominator:
\[ 2a(s-a) + 2c(s-b) = s(2a + 2c - 3b) \]

Expanding both sides:
\[ 2as - 2a^2 + 2cs - 2bc = 2as + 2cs - 3bs \]

### Step 3: Rearranging the equation
Now, we can rearrange the terms:
\[ -2a^2 - 2bc + 3bs = 0 \]

### Step 4: Factor the equation
Rearranging gives:
\[ 2a^2 + 2bc = 3bs \]
This can be rewritten as:
\[ 2a^2 = 3bs - 2bc \]

### Step 5: Analyze the relationship
Now, we can analyze the relationship between \( a, b, c \).

From the equation \( 2a^2 = 3bs - 2bc \), we can derive:
\[ a^2 = \frac{3bs - 2bc}{2} \]

### Step 6: Check for arithmetic progression
To check if \( a, b, c \) are in arithmetic progression, we need to verify if:
\[ b - a = c - b \]
This can be rearranged to:
\[ a + c = 2b \]

### Conclusion
From our derived relationship, we can conclude that:
\[ A + C = 2B \]
Thus, the sides \( a, b, c \) are in Arithmetic Progression (AP).

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

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