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 value of \( P_1 \times P_2 \) where:

\[ P_1 = 1 - \frac{\omega}{2} + \frac{\omega^2}{4} - \frac{\omega^3}{8} + \ldots \]
\[ P_2 = \frac{1 - \omega^2}{2} \]

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

### Step 1: Finding the value of \( \omega \)

The roots of the equation \( x^3 - 1 = 0 \) can be found by factoring:

\[ x^3 - 1 = (x - 1)(x^2 + x + 1) = 0 \]

The real root is \( x = 1 \). The non-real roots are found by solving \( x^2 + x + 1 = 0 \).

Using the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} = \frac{-1 \pm \sqrt{-3}}{2} = \frac{-1 \pm i\sqrt{3}}{2} \]

Thus, the non-real roots are:

\[ \omega = \frac{-1 + i\sqrt{3}}{2}, \quad \omega^2 = \frac{-1 - i\sqrt{3}}{2} \]

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

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

The sum of an infinite geometric series is given by:

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

Substituting the values:

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

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

Using the property \( 1 + \omega + \omega^2 = 0 \), we can express \( 2 + \omega \) as:

\[ 2 + \omega = 1 - \omega^2 \]

Thus, we can rewrite \( P_1 \):

\[ P_1 = \frac{2}{1 - \omega^2} \]

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

Given:

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

### Step 5: Calculating \( P_1 \times P_2 \)

Now we can find \( P_1 \times P_2 \):

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

The \( 2 \) in the numerator and denominator cancels out, and we have:

\[ P_1 \times P_2 = \frac{1 - \omega^2}{1 - \omega^2} = 1 \]

### Final Answer

Thus, the value of \( P_1 \times P_2 \) is:

\[ \boxed{1} \]