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Infinite radical with sixes

Question

Evaluate the infinitely nested expression 6+6+6+⋯\sqrt{6+\sqrt{6+\sqrt{6+\cdots}}}.

Short answer

33. Calling the value xx, self-similarity gives x=6+xx=\sqrt{6+x}, so x2−x−6=(x−3)(x+2)=0x^2-x-6=(x-3)(x+2)=0. The expression is positive, so x=3x=3.

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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