Infinite nested radical value: Evaluate √(30 + √(30 + √(30 + …))) to its exact finite value.
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A5
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B3√10
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C6
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D7
Answer
Correct Answer: 6
Explanation
Introduction / Context:Infinite nested radicals often converge to a finite value x that satisfies a self-referential equation: x = √(constant + x). Solving the resulting quadratic yields the limit value. Only the positive root is meaningful because the radical is nonnegative.Given Data / Assumptions:
- Expression: x = √(30 + x).
- x ≥ 0 due to square roots.
Concept / Approach:Square both sides to eliminate the outer radical and form a quadratic in x. Solve and select the nonnegative solution consistent with the original expression.
Step-by-Step Solution:
Assume limit x exists with x = √(30 + x).Square: x^2 = 30 + x ⇒ x^2 − x − 30 = 0.Solve: Discriminant D = 1 + 120 = 121 ⇒ √D = 11.x = [1 ± 11]/2 ⇒ x = 6 or x = −5.Reject negative root. Hence x = 6.Verification / Alternative check:Plug back: √(30 + 6) = √36 = 6, confirming consistency.
Why Other Options Are Wrong:
- 5, 7: Do not satisfy x = √(30 + x).
- 3√10 ≈ 9.486..., far from the correct fixed point.
Common Pitfalls:Keeping both quadratic roots without checking the radical’s nonnegativity constraint, or mishandling the square step.
Final Answer:
6