Hello.

I have this recurring function or whatever you call it:

$\displaystyle \[\begin{array}{l}

f(0) = 0\\

f(n) = \sqrt {5 + f(n - 1)}

\end{array}\]$

Once $\displaystyle n$ increases, we have $\displaystyle \[\sqrt 5 \]$, inside of which there is another $\displaystyle \[\sqrt 5 \]$ added and so forth (this clearly converges).

As the title says, I am looking forward to obtaining $\displaystyle \[\mathop {\lim }\limits_{n \to \infty } f(n)\]$

P.S. I started from the greatest depth and noticed $\displaystyle \[\sqrt {5 + \sqrt 5 } = \frac{{\sqrt {20 + 4\sqrt 5 } }}{2} = \frac{{2 + \sqrt 5 }}{2}\]$, however later on I am not able to control the expression...

P.P.S. Messing around I've got a good feeling that such infinite summations $\displaystyle \[\sqrt {a + \sqrt {a + \sqrt {a + ...\sqrt a } } } = \frac{{1 + \sqrt {1 + 4a} }}{2}\]$ can be defined like that, no clue how to prove it though