Question

If \(x^2+x+1=0\), then the value of \(\left(x + \frac{1}{x}\right)^2 + \left(x^2 + \frac{1}{x^2}\right)^2 + \left(x^3 + \frac{1}{x^3}\right)^2 + \dots + \left(x^{25} + \frac{1}{x^{25}}\right)^2\) is:

• A. 128
• B. 175
• C. 145
• D. 162

Hint:

Whenever you see the equation \(x^2+x+1=0\), immediately think of the complex cube roots of unity, \(\omega\) and \(\omega^2\).
Their properties (\(\omega^3=1, 1+\omega+\omega^2=0\)) are fundamental to solving such problems quickly.
The value of \(x^n + 1/x^n\) follows a simple periodic pattern.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We are given the equation \(x^2+x+1=0\), whose roots are the complex cube roots of unity, \(\omega\) and \(\omega^2\).
We need to find the sum of the squares of expressions of the form \(x^n + \frac{1}{x^n}\) for \(n\) from 1 to 25.
Step 2: Key Formula or Approach:
The roots of \(x^2+x+1=0\) are \(x = \omega\) and \(x = \omega^2\), where \(\omega = e^{i2\pi/3}\).
Key properties of \(\omega\):
1. \(\omega^3 = 1\)
2. \(1+\omega+\omega^2 = 0\)
3. \(\frac{1}{\omega} = \omega^2\) and \(\frac{1}{\omega^2} = \omega\)
We need to evaluate the term \(S_n = x^n + \frac{1}{x^n}\). Let's take \(x = \omega\).
\(S_n = \omega^n + \frac{1}{\omega^n} = \omega^n + (\omega^2)^n = \omega^n + \omega^{2n}\).
Step 3: Detailed Explanation:
Let's analyze the value of \(S_n = \omega^n + \omega^{2n}\) based on \(n\):
Case 1: n is a multiple of 3.
Let \(n=3k\) for some integer \(k\).
\(S_{3k} = \omega^{3k} + \omega^{6k} = (\omega^3)^k + (\omega^3)^{2k} = 1^k + 1^{2k} = 1+1=2\).
Case 2: n is not a multiple of 3.
In this case, \(n = 3k+1\) or \(n = 3k+2\).
From the property \(1+\omega^n+\omega^{2n}=0\) when \(n\) is not a multiple of 3, we have \(\omega^n+\omega^{2n} = -1\).
So, \(S_n = -1\) if \(n\) is not a multiple of 3.
Now we need to calculate the sum: \(\sum_{n=1}^{25} (S_n)^2\).
We need to find how many numbers from 1 to 25 are multiples of 3.
These are 3, 6, 9, 12, 15, 18, 21, 24. There are 8 such numbers.
The number of terms that are not multiples of 3 is \(25 - 8 = 17\).
The sum can be split into two parts:
\[ \text{Sum} = \sum_{n \text{ is mult of 3}} (S_n)^2 + \sum_{n \text{ is not mult of 3}} (S_n)^2 \] \[ \text{Sum} = (\text{Number of multiples of 3}) \times (2)^2 + (\text{Number of non-multiples of 3}) \times (-1)^2 \] \[ \text{Sum} = 8 \times 4 + 17 \times 1 \] \[ \text{Sum} = 32 + 17 = 49 \] Step 4: Final Answer:
The calculated value of the sum is 49. Since this is not among the options, the question or the provided options are likely incorrect.