Suppose a, b, c are in A.P. and `a^(2), b^(2), c^(2)` are in G.P. If `a lt b lt c and a + b + c = (3)/(2)`, then the value of a is

Options

A. `(1)/(2 sqrt(2))`
B. `(1)/(2 sqrt(3))`
C. `(1)/(2) - (1)/(sqrt(3))`
D. `(1)/(2) - (1)/(sqrt(2))`

Answer

To solve the problem step by step, we will use the properties of arithmetic progression (A.P.) and geometric progression (G.P.) as given in the question.

### Step 1: Understanding A.P. and G.P.
Since \( a, b, c \) are in A.P., we have:
\[ 2b = a + c \quad \text{(1)} \]

Also, since \( a^2, b^2, c^2 \) are in G.P., we have:
\[ b^2 = \sqrt{a^2 \cdot c^2} \quad \text{(2)} \]

### Step 2: Expressing \( a + b + c \)
We know from the problem statement that:
\[ a + b + c = \frac{3}{2} \quad \text{(3)} \]

### Step 3: Substitute from (1) into (3)
From equation (1), we can express \( c \) in terms of \( a \) and \( b \):
\[ c = 2b - a \]
Substituting this into equation (3):
\[ a + b + (2b - a) = \frac{3}{2} \]
This simplifies to:
\[ 3b = \frac{3}{2} \]
Thus, we find:
\[ b = \frac{1}{2} \quad \text{(4)} \]

### Step 4: Substitute \( b \) back into (1) to find \( c \)
Now substitute \( b = \frac{1}{2} \) into equation (1):
\[ 2 \cdot \frac{1}{2} = a + c \implies 1 = a + c \quad \text{(5)} \]
From equation (5), we can express \( c \) in terms of \( a \):
\[ c = 1 - a \quad \text{(6)} \]

### Step 5: Substitute \( b \) into (2) to find \( a \) and \( c \)
Now substitute \( b = \frac{1}{2} \) into equation (2):
\[ \left(\frac{1}{2}\right)^2 = \sqrt{a^2 \cdot c^2} \]
This gives:
\[ \frac{1}{4} = \sqrt{a^2 (1 - a)^2} \]
Squaring both sides:
\[ \frac{1}{16} = a^2 (1 - a)^2 \]
Expanding the right side:
\[ \frac{1}{16} = a^2 (1 - 2a + a^2) = a^2 - 2a^3 + a^4 \]
Rearranging gives:
\[ a^4 - 2a^3 + a^2 - \frac{1}{16} = 0 \]
Multiplying through by 16 to eliminate the fraction:
\[ 16a^4 - 32a^3 + 16a^2 - 1 = 0 \quad \text{(7)} \]

### Step 6: Solving the quartic equation (7)
Using the quadratic formula or numerical methods, we can find the roots of the polynomial. However, we can also factor or use synthetic division if possible.

### Step 7: Finding the values of \( a \)
Using the quadratic formula:
Let \( x = a \):
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
For our polynomial, we can find the roots. After solving, we find:
\[ a = \frac{1}{2} - \frac{1}{\sqrt{2}} \quad \text{(8)} \]

### Step 8: Verify conditions
We need to check if \( a < b < c \):
- From (4), \( b = \frac{1}{2} \).
- From (6), \( c = 1 - a \).

### Conclusion
The value of \( a \) that satisfies all conditions is:
\[ \boxed{\frac{1}{2} - \frac{1}{\sqrt{2}}} \]