If ` x + (1)/( x) = 2 " then " x^( 2013) + (1)/( x^(2014)) ` = ?

Options

A. 0
B. 1
C. `-1`
D. 2

Answer

To solve the problem, we start with the given equation:

**Step 1:** Given \( x + \frac{1}{x} = 2 \).

**Hint:** Recognize that this equation can help us find the value of \( x \).

**Step 2:** We can rearrange the equation to find \( x \). Let's isolate \( x \) by multiplying both sides by \( x \):

\[ x^2 + 1 = 2x \]

**Hint:** Rearranging the equation helps to form a quadratic equation.

**Step 3:** Rearranging gives us:

\[ x^2 - 2x + 1 = 0 \]

**Hint:** This is a standard quadratic equation format.

**Step 4:** We can factor the quadratic equation:

\[ (x - 1)^2 = 0 \]

**Hint:** Look for perfect square factors to simplify the equation.

**Step 5:** Solving this gives:

\[ x - 1 = 0 \implies x = 1 \]

**Hint:** The solution to the equation gives the value of \( x \).

**Step 6:** Now, we need to find \( x^{2013} + \frac{1}{x^{2014}} \). Substituting \( x = 1 \):

\[ x^{2013} + \frac{1}{x^{2014}} = 1^{2013} + \frac{1}{1^{2014}} \]

**Hint:** Remember that any number raised to a power of 1 is still 1.

**Step 7:** This simplifies to:

\[ 1 + 1 = 2 \]

**Hint:** Adding the results gives the final answer.

**Final Answer:** Therefore, \( x^{2013} + \frac{1}{x^{2014}} = 2 \).

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