Find a general formula for $c_{n+1}=c_n+\frac1{c_n}$ (Recursive) Find a general formula for $c_{n+1}$
If $c_1 = 1$ and $$c_{n+1} = c_n + \frac{1}{c_n}$$
$$c_1 = 1$$
$$c_2 = 1 + 1 = \frac{2}{1}$$
$$c_3 = 2 + 1/2 = \frac{5}{2}$$
$$c_4 = \frac{5}{2} + \frac{2}{5} = \frac{29}{10}$$
$$c_5 = \frac{29}{10} + \frac{10}{29} = \frac{941}{290}$$

Searching the Online Encyclopedia of Integer Sequences is always a good idea.
