The value of ` sum _(n = 1) ^(5) ( x ^(n) + (1)/( x^(n)))^(2) " where " x^(2) - x + 1 = 0 ` is

Options

A. 0
B. 10
C. 12
D. none

Answer

To solve the problem, we need to evaluate the expression:

\[ \sum_{n=1}^{5} \left( x^n + \frac{1}{x^n} \right)^2 \]

given that \( x^2 - x + 1 = 0 \).

### Step 1: Find the roots of the equation \( x^2 - x + 1 = 0 \)

Using the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):

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

Thus, the roots are:

\[ x_1 = \frac{1 + \sqrt{3}i}{2}, \quad x_2 = \frac{1 - \sqrt{3}i}{2} \]

### Step 2: Choose one root and express it in exponential form

Let's choose \( x_1 = \frac{1 + \sqrt{3}i}{2} \). We can express this in polar form:

\[ x_1 = \cos\left(\frac{\pi}{3}\right) + i \sin\left(\frac{\pi}{3}\right) = e^{i\frac{\pi}{3}} \]

### Step 3: Calculate \( x^n + \frac{1}{x^n} \)

Using \( x = e^{i\frac{\pi}{3}} \):

\[ x^n = e^{i\frac{n\pi}{3}}, \quad \frac{1}{x^n} = e^{-i\frac{n\pi}{3}} \]

Thus,

\[ x^n + \frac{1}{x^n} = e^{i\frac{n\pi}{3}} + e^{-i\frac{n\pi}{3}} = 2\cos\left(\frac{n\pi}{3}\right) \]

### Step 4: Substitute into the summation

Now we substitute this back into the summation:

\[ \sum_{n=1}^{5} \left( x^n + \frac{1}{x^n} \right)^2 = \sum_{n=1}^{5} \left( 2\cos\left(\frac{n\pi}{3}\right) \right)^2 = 4\sum_{n=1}^{5} \cos^2\left(\frac{n\pi}{3}\right) \]

### Step 5: Calculate \( \cos^2\left(\frac{n\pi}{3}\right) \) for \( n = 1, 2, 3, 4, 5 \)

Calculating the values:

- For \( n=1 \): \( \cos^2\left(\frac{\pi}{3}\right) = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \)
- For \( n=2 \): \( \cos^2\left(\frac{2\pi}{3}\right) = \left(-\frac{1}{2}\right)^2 = \frac{1}{4} \)
- For \( n=3 \): \( \cos^2\left(\pi\right) = (-1)^2 = 1 \)
- For \( n=4 \): \( \cos^2\left(\frac{4\pi}{3}\right) = \left(-\frac{1}{2}\right)^2 = \frac{1}{4} \)
- For \( n=5 \): \( \cos^2\left(\frac{5\pi}{3}\right) = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \)

### Step 6: Sum the values

Now we sum these values:

\[ \sum_{n=1}^{5} \cos^2\left(\frac{n\pi}{3}\right) = \frac{1}{4} + \frac{1}{4} + 1 + \frac{1}{4} + \frac{1}{4} = \frac{5}{4} + 1 = \frac{9}{4} \]

### Step 7: Multiply by 4

Finally, we multiply by 4:

\[ 4 \sum_{n=1}^{5} \cos^2\left(\frac{n\pi}{3}\right) = 4 \cdot \frac{9}{4} = 9 \]

### Final Answer

Thus, the value of the summation is:

\[ \boxed{9} \]