Understanding the Equation
The given equation is:
\[ x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 = 0. \]
This can be rewritten using the formula for the sum of a geometric series. The equation can be factored as:
\[ \frac{x^7 - 1}{x - 1} = 0 \]
for \( x \neq 1 \). This implies that:
\[ x^7 = 1. \]
Thus, the solutions are the seventh roots of unity, excluding \( x = 1 \).
Identifying the Roots
The seventh roots of unity are:
\[ x = e^{2\pi i k / 7}, \]
where \( k = 0, 1, 2, \ldots, 6 \). The relevant roots for our equation are:
- \( x_1 = e^{2\pi i / 7} \)
- \( x_2 = e^{4\pi i / 7} \)
- \( x_3 = e^{6\pi i / 7} \)
- \( x_4 = e^{8\pi i / 7} \)
- \( x_5 = e^{10\pi i / 7} \)
- \( x_6 = e^{12\pi i / 7} \)
Calculating \( x^{5054} + x^{6055} - 7 \)
To solve for \( x^{5054} \) and \( x^{6055} \):
- We find the exponents modulo 7:
- \( 5054 \mod 7 = 3 \)
- \( 6055 \mod 7 = 4 \)
Thus, we have:
\[ x^{5054} = x^3 \]
\[ x^{6055} = x^4 \]
The expression simplifies to:
\[ x^3 + x^4 - 7. \]
Using the original equation \( x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 = 0 \):
Rearranging gives:
\[ x^3 + x^4 = - (x^2 + x + 1 + x^5 + x^6). \]
Since \( x^3 + x^4 + x^5 + x^6 + x^2 + x + 1 = 0 \), we deduce:
\[ x^3 + x^4 = - (x^2 + x + 1). \]
Thus, substituting yields:
\[ - (x^2 + x + 1) - 7 = -7 - 1 = -8. \]
Finally, we arrive at:
\[ -8 + 1 = -5. \]
Final Answer
Hence, the value of \( x^{5054} + x^{6055} - 7 \) is:
Option D: -5.