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

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

Uniteasy Admin · Accepted Answer

Let 
$$x = \sqrt{3+\sqrt{3+\sqrt{3+\sqrt{3+\cdots \infty } } } }$$c 
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
$$x = \sqrt{3+x} $$c
Then,
$$x > 0$$c
Square both sides results in a quadratic equation
$$x^2-x-3=0$$c
Using the root formula for a quadratic equation, we get
$$ x = \dfrac{1\pm\sqrt{1^2-4\cdot 1\cdot (-3)} }{2}  = \dfrac{1\pm\sqrt{13} }{2} $$c
Cancel the negative solution, then,
$$x = \dfrac{1+\sqrt{13} }{2} $$c 
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

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