Find a Closed Form for this Recursively Defined Sequence

Question

Let $\{a_n\}_{1}^{\infty}$ be defined by $a_{1} = 1$ and $a_{n} = \sqrt{1+a_{n-1}}$. The limit of this sequence can easily be determined to exist and subsequently found to be $\phi$ which is suprising to me. I don't see any intuition from the outset as to why this sequence would converge to this known constant. For fun, and for further understanding, I would like to try and find a closed form for this sequence. I am aware that there is a wealth of theory around recurrence relations that I do not have exposure to, so I suspect someone should be able to help with this.