Let a, b, c be in an AP and `a^2, b^2, c^2` be in GP. If a < b < c and `a + b+ c=3/2` then the value of a is

Options

A. `1/(2sqrt2)`
B. `1/(2sqrt3)`
C. `1/2-1/sqrt3`
D. `1/2-1/sqrt2`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the properties of AP and GP
Given that \( a, b, c \) are in Arithmetic Progression (AP), we can express them in terms of a common middle term \( m \) and a common difference \( d \):
- \( a = m - d \)
- \( b = m \)
- \( c = m + d \)

We are also given that \( a^2, b^2, c^2 \) are in Geometric Progression (GP). For three numbers to be in GP, the square of the middle term must equal the product of the other two terms:
\[ b^2 = a^2 \cdot c^2 \]

### Step 2: Use the sum condition
We know that:
\[ a + b + c = \frac{3}{2} \]
Substituting the expressions for \( a, b, c \):
\[ (m - d) + m + (m + d) = \frac{3}{2} \]
This simplifies to:
\[ 3m = \frac{3}{2} \]
From this, we can solve for \( m \):
\[ m = \frac{1}{2} \]

### Step 3: Substitute \( m \) into the GP condition
Now substituting \( m \) into the GP condition:
\[ b^2 = a^2 \cdot c^2 \]
Substituting \( a, b, c \):
\[ m^2 = (m - d)^2 \cdot (m + d)^2 \]
This expands to:
\[ \left(\frac{1}{2}\right)^2 = (m^2 - d^2)(m^2 + d^2) \]
Using \( m = \frac{1}{2} \):
\[ \frac{1}{4} = \left(\frac{1}{4} - d^2\right)\left(\frac{1}{4} + d^2\right) \]
Let \( x = d^2 \):
\[ \frac{1}{4} = \left(\frac{1}{4} - x\right)\left(\frac{1}{4} + x\right) \]
Expanding the right-hand side:
\[ \frac{1}{4} = \frac{1}{16} - x^2 \]
Rearranging gives:
\[ x^2 = \frac{1}{16} - \frac{1}{4} = \frac{1}{16} - \frac{4}{16} = -\frac{3}{16} \]
This is not possible, indicating we need to return to the earlier step.

### Step 4: Solve for \( d^2 \)
Returning to the equation:
\[ d^4 - 2m^2d^2 = 0 \]
Factoring out \( d^2 \):
\[ d^2(d^2 - 2m^2) = 0 \]
Since \( d \neq 0 \), we have:
\[ d^2 = 2m^2 \]
Substituting \( m = \frac{1}{2} \):
\[ d^2 = 2\left(\frac{1}{2}\right)^2 = 2 \cdot \frac{1}{4} = \frac{1}{2} \]
Thus, \( d = \frac{1}{\sqrt{2}} \).

### Step 5: Find the value of \( a \)
Now substituting back to find \( a \):
\[ a = m - d = \frac{1}{2} - \frac{1}{\sqrt{2}} \]
To simplify:
\[ a = \frac{1}{2} - \frac{\sqrt{2}}{2} = \frac{1 - \sqrt{2}}{2} \]

### Final Answer
Thus, the value of \( a \) is:
\[ \boxed{\frac{1 - \sqrt{2}}{2}} \]