Question

If \( x + \frac{1}{x} = 2 \), find the value of \( x^{10} + \frac{1}{x^{10}} \).

• A. 1024
• B. 2
• C. 1
• D. 0

Hint:

Use algebraic identity recurrence: \( A_n = a A_{n-1} - A_{n-2} \) where \( A_n = x^n + \frac{1}{x^n} \)

Answer 1

The Correct Option is B

Given: \[ x + \frac{1}{x} = 2 \] This is only possible if \( x = 1 \), since: \[ x + \frac{1}{x} = 2 \Rightarrow x = 1 \Rightarrow x^{10} + \frac{1}{x^{10}} = 1 + 1 = \boxed{2} \] Alternative method (sequence identity): Let \( A_n = x^n + \frac{1}{x^n} \) Use identity: \[ A_n = (x + \frac{1}{x}) A_{n-1} - A_{n-2} \Rightarrow A_1 = 2, A_0 = 2, A_2 = 2^2 - 2 = 2 \Rightarrow A_3 = 2 A_2 - A_1 = 4 - 2 = 2 \Rightarrow \text{All } A_n = 2 \Rightarrow A_{10} = \boxed{2} \]