Is this element in $\ell_2$?

Is this element in $\ell_2$?

Let $(y_1,y_2,\cdots, y_n,\cdots)$ be an element of $\ell_2$. Is it true
that for any complex number $\lambda$ , the sequence defined recursively
by $$x_1=\lambda y_1, x_n=\lambda\left(y_n+\frac{x_{n-1}}{n-1}\right),
n=2,3,\cdots$$ is also in $\ell_2$?