sequences and series - Recursive square root inside square root problem

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!