A purely verification proof of the first Rogers-Ramanujan identity (Q914679)

It is known that the first Rogers-Ramanujan identity follows from the following Andrews identity: \[ \sum_{k}\frac{q^{k^ 2}}{(q)_ k(q)_{n-k}}=\sum_{k}\frac{(-1)^ kq^{(5k^ 2-k)/2}}{(q)_{n- k}(q)_{n+k}}. \] The proof of the identity given here involves showing that both sides are solutions of a certain fifth order linear recurrence.