Finding resolvent of an operator

Question

Let $A:l^p\to l^p$ be defined as $Ax=(\frac{n}{n+1}x_{n+1})_{n\geq 1}$, for $p=2$ and $p=\infty$. I have to deduce resolvent, or to solve $(A-\lambda I)x=y$, but I got $\frac{n}{n+1}x_{n+1}-\lambda x_n=y_n$ and don't know find x from this equation. Any help is welcome. Thanks in advance.