Let $x = 1 + \frac{1}{1 + \frac{1}{1 + \dots \infty}}$. Which of the following is correct?

### Concept & Strategy This problem involves an **Infinite Continued Fraction**. The core strategy relies on the principle of self-similarity: because the fraction goes on infinitely, a nested portion of it is identical to the whole expression. We can substitute $x$ back into its own equation to form a standard quadratic equation. ### Step-by-Step Solution * **Step 1: Identify the repeating structure** We are given: $$x = 1 + \frac{1}{1 + \frac{1}{1 + \dots \infty}}$$ * **Step 2: Apply self-similarity** Look closely at the denominator of the first fraction. It is exactly the same infinite series as the original expression for $x$. $$1 + \frac{1}{1 + \dots \infty} = x$$ Therefore, we can substitute $x$ into the denominator: $$x = 1 + \frac{1}{x}$$ * **Step 3: Solve for the quadratic equation** Multiply the entire equation by $x$ to eliminate the fraction: $$x \cdot (x) = x \cdot \left(1 + \frac{1}{x}\right)$$ $$x^2 = x + 1$$ Rearrange the terms to set the equation to zero: $$x^2 - x - 1 = 0$$ ### Exam Strategy & Shortcut Whenever you see a pattern of $x = A + \frac{B}{A + \frac{B}{\dots \infty}}$, you can instantly jump to the equation $x = A + \frac{B}{x}$. For this specific problem where $A=1$ and $B=1$, you should immediately write $x = 1 + 1/x$, which mentally rearranges to $x^2 - x - 1 = 0$ in seconds. ### Common Pitfall The most frequent error is misidentifying where the repeating part begins. Some students might incorrectly substitute $x$ too early or too late, leading to equations like $x = 1/(1+x)$, which represents a different infinite fraction ($x = \frac{1}{1 + \frac{1}{1+\dots}}$). Pay close attention to the leading integer. ### Final Answer Therefore, the correct answer is **$x^2 - x - 1 = 0$**.