Value of infinite nested radical: Evaluate √(6 + √(6 + √(6 + …))) to its exact finite value.

Difficulty: Easy

Correct Answer: 3

Explanation:

Introduction / Context:
Nested radicals of the form √(k + √(k + …)) typically converge to a finite limit x satisfying x = √(k + x). Solving the resulting quadratic yields the exact value, with the positive root chosen due to the square root’s nonnegativity.

Given Data / Assumptions:

Expression: x = √(6 + x).
x ≥ 0.

Concept / Approach:
Square both sides to eliminate the radical and solve the quadratic equation. Only the nonnegative solution is valid for the nested radical limit.

Step-by-Step Solution:

Assume convergence x = √(6 + x).Square: x^2 = 6 + x ⇒ x^2 − x − 6 = 0.Discriminant D = 1 + 24 = 25 ⇒ √D = 5.x = [1 ± 5]/2 ⇒ x = 3 or x = −2.Reject −2 (x must be ≥ 0). Thus x = 3.

Verification / Alternative check:
Check: √(6 + 3) = √9 = 3, consistent. Iterating numerically from any positive seed also converges near 3.

Why Other Options Are Wrong:

2, 4, 5: None satisfy x = √(6 + x).

Common Pitfalls:
Keeping both quadratic roots or forgetting that the expression must be nonnegative. Also, confusing this with finite-depth radicals, which do not satisfy the same fixed-point equation.

Value of infinite nested radical: Evaluate √(6 + √(6 + √(6 + …))) to its exact finite value.

More Questions from Quadratic Equation

Discussion & Comments