Code in python using import sympy$.$

Given the sequence defined recursively via

$a_1=1,$ and $,a_{n+1}=1+\frac{1}{1+a_n}$

$($a$)$ Find the first $10$ terms of the sequence $($as approximate decimals$)$ and guess the value of the limit based on these values.

$($b$)$ Plot the first $50$ terms of the sequence and guess the value of the limit based on the graph.

$($c$)$ Find the limit of this recursive sequence.