Question

If $a + 1/a =2$ what is $a^{2024}+\frac{1}{a^{2024}}=?$

• A. 2
• B. $$2^{2024}$$
• C. 2024
• D. $$2024^2$$

Answer 1

Answer:

A. 2

Understanding the Equation

The given equation is $a + 1/a = 2$. This is a fundamental algebraic relationship that can be simplified.

Solving for 'a'

• Multiply the entire equation by 'a' to eliminate the fraction: $a(a + 1/a) = 2a$

• This simplifies to: $a^2 + 1 = 2a$

• Rearrange the terms to form a quadratic equation: $a^2 - 2a + 1 = 0$

• This quadratic equation is a perfect square trinomial: $(a - 1)^2 = 0$

• Taking the square root of both sides gives: $a - 1 = 0$

• Therefore, the value of 'a' is $a = 1$.

Applying the Value of 'a'

Now that we know $a = 1$, we can substitute this value into the expression we need to evaluate: $a^{2024} + \frac{1}{a^{2024}}$.

Calculation

• Substitute $a=1$: $1^{2024} + \frac{1}{1^{2024}}$

• Any positive integer power of 1 is 1: $1 + \frac{1}{1}$

• Simplify the expression: $1 + 1$

• The final result is $2$.