#### Question

Find the value of \sqrt{3+\sqrt{3+\sqrt{3+\sqrt{3+\cdots \infty } } } } when the layers of the square root go to infinite

Question

Let

x = \sqrt{3+\sqrt{3+\sqrt{3+\sqrt{3+\cdots \infty } } } }

when the layers of square root go to infinite

which makes no difference starting to count from the second square root.

Then, we get the following equation

x = \sqrt{3+x}

Then,

x > 0

Square both sides results in a quadratic equation

x^2-x-3=0

Using the root formula for a quadratic equation, we get

x = \dfrac{1\pm\sqrt{1^2-4\cdot 1\cdot (-3)} }{2} = \dfrac{1\pm\sqrt{13} }{2}

Cancel the negative solution, then,

x = \dfrac{1+\sqrt{13} }{2}

In summary,

the value of \sqrt{3+\sqrt{3+\sqrt{3+\sqrt{3+\cdots \infty } } } } = \dfrac{1+\sqrt{13} }{2} when the layers of the square root go to infinite