Nested Square Roots 2

Find the value of \sqrt{16+\sqrt{16+\sqrt{16+\cdots}}}
Source: NCTM Mathematics Teacher, October 2006

Solution
Let x=\sqrt{16+\sqrt{16+\sqrt{16+\cdots}}}
By substitution
x=\sqrt{16+x}
Squaring both sides
x^2=16+x
x^2-x-16=0
Using the quadratic formula to solve for x
x=\dfrac{1\pm\sqrt{65}}{2}
\dfrac{1-\sqrt{65}}{2} is an extraneous solution because the expression cannot be negative.
\sqrt{16+\sqrt{16+\sqrt{16+\cdots}}}=\dfrac{1+\sqrt{65}}{2}

Answer: \dfrac{1+\sqrt{65}}{2}

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About mvtrinh

Retired high school math teacher.
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