If `z^(2)+z+1=0` then the value of
`(z+1/z)^(2)+(z^(2)+1/z^(2))^(2)+(z^(3)+1/z^(3))^(2)+....+(z^(21)+1/z^(21))^(2)` is equal to

Options

A. 21
B. 42
C. 0
D. 11

Answer

To solve the equation \( z^2 + z + 1 = 0 \) and find the value of

\[ (z + \frac{1}{z})^2 + (z^2 + \frac{1}{z^2})^2 + (z^3 + \frac{1}{z^3})^2 + \ldots + (z^{21} + \frac{1}{z^{21}})^2, \]

we will proceed step by step.

### Step 1: Find the roots of the quadratic equation

The roots of the equation \( z^2 + z + 1 = 0 \) can be found using the quadratic formula:

\[ z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Here, \( a = 1, b = 1, c = 1 \). Plugging in these values:

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

Thus, the roots are:

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

### Step 2: Use properties of roots of unity

The roots \( \omega \) and \( \omega^2 \) are the non-real cube roots of unity. They satisfy:

\[ \omega^3 = 1 \quad \text{and} \quad 1 + \omega + \omega^2 = 0 \]

### Step 3: Calculate \( z + \frac{1}{z} \)

We can express \( \frac{1}{z} \) as follows:

\[ \frac{1}{z} = \frac{1}{\omega} = \omega^2 \]

Thus,

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

### Step 4: Calculate \( (z + \frac{1}{z})^2 \)

Now we compute:

\[ (z + \frac{1}{z})^2 = (-1)^2 = 1 \]

### Step 5: Calculate \( z^2 + \frac{1}{z^2} \)

Using the identity:

\[ z^2 + \frac{1}{z^2} = (z + \frac{1}{z})^2 - 2 \]

we have:

\[ z^2 + \frac{1}{z^2} = 1 - 2 = -1 \]

### Step 6: Calculate \( (z^2 + \frac{1}{z^2})^2 \)

Now we compute:

\[ (z^2 + \frac{1}{z^2})^2 = (-1)^2 = 1 \]

### Step 7: Calculate \( z^3 + \frac{1}{z^3} \)

Using the identity:

\[ z^3 + \frac{1}{z^3} = (z + \frac{1}{z})(z^2 + \frac{1}{z^2}) - (z + \frac{1}{z}) \]

we find:

\[ z^3 + \frac{1}{z^3} = (-1)(-1) - (-1) = 1 + 1 = 2 \]

### Step 8: Calculate \( (z^3 + \frac{1}{z^3})^2 \)

Now we compute:

\[ (z^3 + \frac{1}{z^3})^2 = 2^2 = 4 \]

### Step 9: Identify the pattern

The terms \( z^n + \frac{1}{z^n} \) will repeat every three terms due to the properties of cube roots of unity. Thus, we have:

\[ (z + \frac{1}{z})^2 = 1, \quad (z^2 + \frac{1}{z^2})^2 = 1, \quad (z^3 + \frac{1}{z^3})^2 = 4 \]

### Step 10: Calculate the total sum

Since there are 21 terms, we can group them into sets of 3:

\[ \text{Number of complete sets} = \frac{21}{3} = 7 \]

The total sum is:

\[ 7 \times (1 + 1 + 4) = 7 \times 6 = 42 \]

### Final Answer

Thus, the value of the expression is:

\[ \boxed{42} \]