If $x = \sqrt{1 + \sqrt{1 + \sqrt{1 + \cdots \infty}}}$, then the positive value of $x$ is

### Concept & Formula This problem involves an infinite nested square root sequence. The standard approach is to use substitution to form a quadratic equation. The general formula for an infinite series of the form $\sqrt{a + \sqrt{a + \sqrt{a + \cdots}}}$ is: $$ x = \frac{\sqrt{4a + 1} + 1}{2} $$ ### Step-by-Step Solution * **Given:** $x = \sqrt{1 + \sqrt{1 + \sqrt{1 + \cdots \infty}}}$ * Since the sequence goes on infinitely, the expression inside the outermost square root after the "1 +" is identical to the original expression $x$. * Substitute $x$ back into the equation: * $x = \sqrt{1 + x}$ * Square both sides to eliminate the square root: * $x^2 = 1 + x$ * Rearrange the terms to form a standard quadratic equation ($ax^2 + bx + c = 0$): * $x^2 - x - 1 = 0$ * Apply the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a = 1, b = -1, c = -1$: * $x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-1)}}{2(1)}$ * $x = \frac{1 \pm \sqrt{1 + 4}}{2}$ * $x = \frac{1 \pm \sqrt{5}}{2}$ * The question asks for the positive value. Since $\sqrt{5}$ is greater than 1, $\frac{1 - \sqrt{5}}{2}$ would be negative. * Therefore, we take the positive root. ### Exam Strategy & Shortcut Instead of deriving the quadratic equation every time, memorize the direct shortcut formula for any infinite series $\sqrt{a + \sqrt{a + \cdots}} = \frac{\sqrt{4a + 1} + 1}{2}$. Here, $a = 1$. Plugging it directly into the formula gives $\frac{\sqrt{4(1) + 1} + 1}{2} = \frac{\sqrt{5} + 1}{2}$. You can arrive at the answer in under 5 seconds. ### Common Pitfall A common mistake is trying to guess factors for the number 1 as if it were an easily factorable number like 6 or 12. Since 1 cannot be factored into two consecutive integers, guessing fails, and students often abandon the question or pick a random root instead of applying the quadratic formula. ### Final Answer **Therefore, the correct answer is $\frac{\sqrt{5} + 1}{2}$.**