If ` t^(2) + t + 1 = 0` then, the value of `( t + (1)/( t))^(2) + ( t^(2) + (1)/( t^(2)))^(2) + . . . + (t^(27) + (1)/(t^(27)))^(2)` is

Options

A. 27
B. 72
C. 45
D. 54

Answer

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

\[ (t + \frac{1}{t})^2 + (t^2 + \frac{1}{t^2})^2 + \ldots + (t^{27} + \frac{1}{t^{27}})^2, \]

we can follow these steps:

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

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

Let \( \omega = \frac{-1 + i\sqrt{3}}{2} \) and \( \omega^2 = \frac{-1 - i\sqrt{3}}{2} \). These are the two non-real cube roots of unity.

### Step 2: Properties of \( \omega \)
Since \( \omega^3 = 1 \), we know that:

\[ \omega^0 = 1, \quad \omega^1 = \omega, \quad \omega^2 = \omega^2, \quad \omega^3 = 1, \quad \omega^4 = \omega, \ldots \]

### Step 3: Calculate \( t + \frac{1}{t} \)
We can calculate \( t + \frac{1}{t} \):

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

### Step 4: Calculate \( t^n + \frac{1}{t^n} \)
Using the properties of \( \omega \):

\[ t^n + \frac{1}{t^n} = \omega^n + \omega^{-n}. \]

This expression will yield:

- If \( n \equiv 0 \mod 3 \), then \( t^n + \frac{1}{t^n} = 2 \).
- If \( n \equiv 1 \mod 3 \), then \( t^n + \frac{1}{t^n} = -1 \).
- If \( n \equiv 2 \mod 3 \), then \( t^n + \frac{1}{t^n} = -1 \).

### Step 5: Calculate the squares
Now we need to compute:

\[ (t^n + \frac{1}{t^n})^2. \]

This gives us:

- If \( n \equiv 0 \mod 3 \), then \( (t^n + \frac{1}{t^n})^2 = 4 \).
- If \( n \equiv 1 \mod 3 \) or \( n \equiv 2 \mod 3 \), then \( (t^n + \frac{1}{t^n})^2 = 1 \).

### Step 6: Count occurrences
From \( n = 0 \) to \( n = 27 \):

- The values of \( n \equiv 0 \mod 3 \): \( 0, 3, 6, 9, 12, 15, 18, 21, 24, 27 \) (10 terms).
- The values of \( n \equiv 1 \mod 3 \): \( 1, 4, 7, 10, 13, 16, 19, 22, 25 \) (9 terms).
- The values of \( n \equiv 2 \mod 3 \): \( 2, 5, 8, 11, 14, 17, 20, 23, 26 \) (9 terms).

### Step 7: Calculate the total
Now we can calculate the total:

\[ \text{Total} = 10 \cdot 4 + 9 \cdot 1 + 9 \cdot 1 = 40 + 9 + 9 = 58. \]

### Final Answer
Thus, the value of

\[ (t + \frac{1}{t})^2 + (t^2 + \frac{1}{t^2})^2 + \ldots + (t^{27} + \frac{1}{t^{27}})^2 = 58. \]