Proposition: $\sqrt{x + \sqrt{x + \sqrt{x + ...}}} = \frac{1 + \sqrt{1 + 4x}}{2}$. Proposition: $$\sqrt{x + \sqrt{x + \sqrt{x + ...}}} = \frac{1 + \sqrt{1 + 4x}}{2}$$
I believe that this is true, and, using Desmos Graphing Calculator, it seems to be true.
I will add how I derived the formula in a moment, if you would like.
Working
I will be honest; all that I did was use the Desmos Graphing Calculator, and let $y = \sqrt{x + \sqrt{x + \sqrt{x + ...}}},$ let $x = 1, 2, 3, ...$, looked at the point at which the two graphs meet, and searched for the number in Google.
It turned up an interesting website, which you may access here, which seemed to show a pattern.
I used this pattern to derive the formula that I stated earlier.

it is true because :
$$S=\sqrt { x+\sqrt { x+\sqrt { x+... }  }  } \\ S=\sqrt { x+S } \\ { S }^{ 2 }-S-x=0\\ S=\frac { 1+\sqrt { 1+4x }  }{ 2 } $$
