Recursive square root inside square root problem

Question

I have been debating this issue for days:

I can't find a recursive function of this equation:

$\large{\sqrt{2+\pi \sqrt{3+\pi\sqrt{4+\pi\sqrt{5+\dotsb}}}}}$

has been trying to find a solution this for days now, is what I have achieved so far:

$f(n)=\sqrt{2 f(n-1)}, f(1)=\sqrt{2}$

Unfortunately, I do not know how to move forward, thanks a lot!

Answer 1

I would write it as $$ f\left(n\right)=\sqrt{n+1+\pi f\left(n+1\right)}, $$ Hence $$ f\left(1\right)=\sqrt{2+\pi f\left(2\right)}, \ f\left(2\right)=\sqrt{3+\pi f\left(3\right)} \Rightarrow f\left(1\right)=\sqrt{2+\pi\sqrt{3+f\left(3\right)}} $$ which means you search for $f\left(1\right)$. $$ f\left(n+1\right)=\frac{1}{\pi}\left(f\left(n\right)^2-n-1\right) $$