What are the methods that can be used to solve recurrence relations such as,
$a_{n+1}=a_n+\frac{1}{a_n}$ ,
$a_{n+1}=a_n-\frac{1}{a_n}$ ,
and reduce $a_n$ to a closed-form formula? And are there any general ways to solve this for arbitrary negative powers like $a_{n+1}=a_n+a_n^{-k}$ ?
Taking the first one $$a_{n+1}=a_n+\frac{1}{a_n}\qquad \text{with} \qquad a_1=1$$ have a look here.
No closed form (neither asymptotics).