If `P_(1) = 1 - (w)/2 + (w^2)/4 - (w^3)/(8) + ……… oo` and `P_(2) = (1 - omega^2)/2` { where w is non-real root of equation `x^3 = 1`} , then `P_(1)P_(2)` is equal to

Answer

To solve the problem, we need to find the product \( P_1 P_2 \) where:

1. \( P_1 = 1 - \frac{\omega}{2} + \frac{\omega^2}{4} - \frac{\omega^3}{8} + \ldots \)
2. \( P_2 = \frac{1 - \omega^2}{2} \)

Here, \( \omega \) is a non-real root of the equation \( x^3 = 1 \).

### Step 1: Identify the non-real roots of \( x^3 = 1 \)

The roots of the equation \( x^3 = 1 \) are:
- \( \omega_0 = 1 \)
- \( \omega_1 = \frac{-1 + \sqrt{3}i}{2} \)
- \( \omega_2 = \frac{-1 - \sqrt{3}i}{2} \)

We can choose \( \omega = \frac{-1 + \sqrt{3}i}{2} \) as our non-real root.

### Step 2: Analyze \( P_1 \)

The series \( P_1 \) can be recognized as an infinite geometric series. The first term \( a = 1 \) and the common ratio \( r = -\frac{\omega}{2} \).

The formula for the sum of an infinite geometric series is given by:

\[ S = \frac{a}{1 - r} \]

### Step 3: Calculate \( P_1 \)

Substituting the values of \( a \) and \( r \):

\[ P_1 = \frac{1}{1 - \left(-\frac{\omega}{2}\right)} = \frac{1}{1 + \frac{\omega}{2}} = \frac{2}{2 + \omega} \]

### Step 4: Calculate \( P_2 \)

Now, we calculate \( P_2 \):

\[ P_2 = \frac{1 - \omega^2}{2} \]

### Step 5: Find \( \omega^2 \)

To find \( \omega^2 \):

\[ \omega^2 = \left(\frac{-1 + \sqrt{3}i}{2}\right)^2 = \frac{(-1)^2 + 2(-1)(\sqrt{3}i) + (\sqrt{3}i)^2}{4} = \frac{1 - 2\sqrt{3}i - 3}{4} = \frac{-2 - 2\sqrt{3}i}{4} = \frac{-1 - \sqrt{3}i}{2} \]

### Step 6: Substitute \( \omega^2 \) into \( P_2 \)

Now substituting \( \omega^2 \) into \( P_2 \):

\[ P_2 = \frac{1 - \left(\frac{-1 - \sqrt{3}i}{2}\right)}{2} = \frac{1 + \frac{1 + \sqrt{3}i}{2}}{2} = \frac{\frac{2 + 1 + \sqrt{3}i}{2}}{2} = \frac{3 + \sqrt{3}i}{4} \]

### Step 7: Calculate \( P_1 P_2 \)

Now we can calculate \( P_1 P_2 \):

\[ P_1 P_2 = \left(\frac{2}{2 + \omega}\right) \left(\frac{1 - \omega^2}{2}\right) \]

Substituting \( P_2 \):

\[ P_1 P_2 = \frac{2}{2 + \omega} \cdot \frac{1 - \left(\frac{-1 - \sqrt{3}i}{2}\right)}{2} \]

### Step 8: Simplify \( P_1 P_2 \)

After simplification, we find that:

\[ P_1 P_2 = 1 \]

### Final Answer

Thus, the final result is:

\[ P_1 P_2 = 1 \]