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Compute the value of the expression \(\sqrt{10 + \sqrt{10 + \sqrt{10 + \dotsb}}}\)

 Dec 8, 2019
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\(\text{Let }x = \sqrt{10 +\sqrt{10 +\sqrt{10 + \cdots}}}\\ x = \sqrt{10 + x}, x \geq 0\\ x^2 = 10 + x\\ x^2 - x - 10 = 0\\ x = \dfrac{1\pm\sqrt{1^2 - 4(1)(-10)}}{2}\\ x = \dfrac{1\pm\sqrt{41}}{2}\\ \text{Rejecting the negative root, we have }\boxed{x = \dfrac{1+\sqrt{41}}{2}} \)

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 Dec 8, 2019

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