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Okay, so I need to figure out how to solve this problem: Find the value of the expression √(1 + √(1 + √(1 + ...))). Hmm, infinite nested square roots, right? I remember seeing something like this before, maybe in algebra or calculus. Let me think.


First, let me write down the expression more clearly. Let’s denote the entire infinite nested radical as x. So, x = √(1 + √(1 + √(1 + ...))). Since the pattern repeats infinitely, the part inside the first square root is actually the same as x itself. That means the equation can be written as x = √(1 + x). Is that right? Let me check.


If I substitute the inner part with x, then yes, it becomes x = √(1 + x). That seems like a standard approach for these infinitely recursive expressions. Now, to solve for x, I can square both sides to eliminate the square root. So, squaring both sides gives x² = 1 + x. Then, rearranging the terms, we get x² - x - 1 = 0. That's a quadratic equation.


To solve the quadratic equation x² - x - 1 = 0, I can use the quadratic formula. The quadratic formula states that for ax² + bx + c = 0, the solutions are x = [-b ± √(b² - 4ac)] / (2a). In this case, a = 1, b = -1, and c = -1. Plugging those values in, we get x = [1 ± √(1 + 4)] / 2 = [1 ± √5] / 2. So, the solutions are (1 + √5)/2 and (1 - √5)/2.


But wait, since we're dealing with square roots, the value must be positive. The square root expression is definitely positive because all the terms inside are positive. So, (1 - √5)/2 is a negative number because √5 is approximately 2.236, so 1 - 2.236 is about -1.236, divided by 2 is approximately -0.618. That’s negative, which doesn't make sense here. Therefore, we discard the negative solution and take the positive one, which is (1 + √5)/2.


Let me verify this result. Let's compute (1 + √5)/2. √5 is approximately 2.236, so adding 1 gives 3.236, divided by 2 is approximately 1.618. Now, let's check if this value satisfies the original equation. If x ≈ 1.618, then √(1 + x) ≈ √(1 + 1.618) ≈ √2.618 ≈ 1.618, which matches. So that seems correct.


But wait, is there a way to confirm this without approximating? Let's see. If x = (1 + √5)/2, then x² should equal 1 + x. Let's compute x²:


x² = [(1 + √5)/2]^2 = (1 + 2√5 + 5)/4 = (6 + 2√5)/4 = (3 + √5)/2.


Now, 1 + x = 1 + (1 + √5)/2 = (2 + 1 + √5)/2 = (3 + √5)/2.




Yes, so x² = 1 + x, which confirms that the solution is correct. Therefore, the value of the infinite nested radical is indeed the golden ratio, (1 + √5)/2.


But let me think again if there's any assumption I made here that might be incorrect. The key step was setting x = √(1 + x), assuming that the expression converges. Is convergence guaranteed here? For these types of infinitely nested radicals, convergence usually depends on the terms inside. Since each term is positive and the sequence is monotonically increasing but bounded above, by the Monotone Convergence Theorem, it should converge.


Let me check that. Let’s define the sequence recursively: a₁ = √1 = 1, a₂ = √(1 + a₁) = √2 ≈ 1.414, a₃ = √(1 + a₂) ≈ √(2.414) ≈ 1.553, a₄ ≈ √(1 + 1.553) ≈ √2.553 ≈ 1.598, and so on. Each term is larger than the previous one, so it's increasing. Now, is it bounded above? Suppose the limit is L, which we found to be (1 + √5)/2 ≈ 1.618. Let's check if the sequence is bounded above by L. Assume aₙ < L, then aₙ₊₁ = √(1 + aₙ) < √(1 + L) = L. Since L² = 1 + L, so √(1 + L) = L. Therefore, by induction, if a₁ < L, then all terms are less than L, and the sequence is increasing and bounded above by L, hence convergent. Therefore, the convergence is justified.


Therefore, the solution is valid. So, the value of the expression is (1 + √5)/2, which is the golden ratio. I think that's the final answer. Let me just recap to make sure I didn't skip any steps.


Let x be the value of the infinite nested radical.
Then, x = √(1 + x).
Squaring both sides: x² = 1 + x.
Rearranging: x² - x - 1 = 0.
Solving quadratic: x = [1 ± √5]/2.
Discarding negative solution: x = (1 + √5)/2.
Verified convergence via Monotone Convergence Theorem.


Yes, that all checks out. I think that's solid. No mistakes noticed here. So the answer should be the golden ratio, (1 + √5)/2.




The value of the infinite nested radical expression ( \sqrt{1 + \sqrt{1 + \sqrt{1 + \cdots}}} ) can be determined by setting it equal to ( x ) and solving the resulting equation:


Let ( x = \sqrt{1 + x} ).
Square both sides: ( x^2 = 1 + x ).
Rearrange into a quadratic equation: ( x^2 - x - 1 = 0 ).
Solve using the quadratic formula:

[
x = \frac{1 \pm \sqrt{1 + 4}}{2} = \frac{1 \pm \sqrt{5}}{2}.
]
Since ( x ) must be positive, discard the negative solution:

[
x = \frac{1 + \sqrt{5}}{2}.
]


Verification:


Convergence is ensured as the sequence is monotonically increasing and bounded above.
Substituting ( x = \frac{1 + \sqrt{5}}{2} ) back confirms ( x^2 = 1 + x ).


Final Answer:

[
\boxed{\frac{1 + \sqrt{5}}{2}}
]