Poisson distribution and expectation

Question

(More about Poisson distributions, Part I):

Let $X$ be a random variable with Poisson $\lambda$ distribution, $\lambda> 0$ (I): Show that $E [X]k = \lambda k$ for any $k$ (- N0. (hint: $X_n= X(X-1)(X-2).........(X-k+1)$)

(II) Calculate $E = E((X-\lambda\sqrt{\lambda)})b$ for $b = 1$, $2$, $3$