I am trying to evaluate $$ \sqrt{4+\sqrt{8+\sqrt{32+\sqrt{512+\sqrt{\frac{512^2}{2}+\sqrt{...}}}}}} $$ where those numbers inside roots are $$ a_{n+1}=\frac{a_n^2}{2}$$ And I found two ways to solve it that give different answers. I believe one of those is not right, but I don't know which and why. Please help.
Method-1. $$x+1=\sqrt{x^2+2x+1}=\sqrt{x^2+x+\sqrt{x^2+2x+1}}\\ =\sqrt{x^2+x+\sqrt{x^2+x+\sqrt{x^2+2x+1}}}=...$$ $x=1\rightarrow$ $$2=\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{...}}}}}}$$ Therefore $$\begin{align} &4=2\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{...}}}}}}\\ &=\sqrt{2^2\cdot2+2^2\cdot\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{...}}}}}}\\ &=\sqrt{8+\sqrt{2^4\cdot2+2^4\cdot\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{...}}}}}}\\ &=\sqrt{8+\sqrt{32+\sqrt{512+\sqrt{\frac{512^2}{2}+\sqrt{...}}}}}\end{align}$$ Finally, $$\sqrt{4+4}=\sqrt{4+\sqrt{8+\sqrt{32+\sqrt{512+\sqrt{\frac{512^2}{2}+\sqrt{...}}}}}}=2\sqrt2$$
Method-2. $$\begin{align} &x+2=\sqrt{x^2+4x+4}=\sqrt{x^2+3x+\sqrt{x^2+8x+16}}\\ &=\sqrt{x^2+3x+\sqrt{x^2+7x+\sqrt{x^2+32x+256}}}\\ &=\sqrt{x^2+3x+\sqrt{x^2+7x+\sqrt{x^2+31x+\sqrt{x^2+512x+256^2}}}}... \end{align}$$ $x=1\rightarrow$ $$3=\sqrt{4+\sqrt{8+\sqrt{32+\sqrt{512+\sqrt{\frac{512^2}{2}+\sqrt{...}}}}}} $$
Alternative Method-2. $$\begin{align} &3=\sqrt9=\sqrt{4+5}=\sqrt{4+\sqrt{25}}=\sqrt{4+\sqrt{8+17}}\\ &=\sqrt{4+\sqrt{8+\sqrt{2\cdot16+16^2+1}}}\\ &=\sqrt{4+\sqrt{8+\sqrt{32+\sqrt{2\cdot16^2+16^4+1}}}}\\ &=\sqrt{4+\sqrt{8+\sqrt{32+\sqrt{512+\sqrt{2\cdot16^4+16^8+1}}}}}\\ &=\sqrt{4+\sqrt{8+\sqrt{32+\sqrt{512+\sqrt{\frac{512^2}{2}+\sqrt{2\cdot16^8+16^{16}+1}}}}}}=... \end{align}$$
So, I have two answers $2\sqrt2$ and $3$. Which one is correct and what's the problem in the other solution?? Thanks.
Now I think I understand. Thanks for all the answers. Let me post this method-3 just to show that it could be any number $\geq2\sqrt2$ and conclude this topic.
Method-3.
$$\begin{align} &\sqrt{10}=\sqrt{4+6}=\sqrt{4+\sqrt{36}}=\sqrt{4+\sqrt{8+28}}\\ &=\sqrt{4+\sqrt{8+\sqrt{32+752}}}\\ &=\sqrt{4+\sqrt{8+\sqrt{32+\sqrt{512+564992}}}}\\ &=\sqrt{4+\sqrt{8+\sqrt{32+\sqrt{512+\sqrt{\frac{512^2}{2}+\sqrt{...}}}}}}=... \end{align}$$