a) Show by counting that

$${b}_{0}=1\text{and}{b}_{n}=k(k-1{)}^{n-1}\text{for}n\ge 1.$$.

b) Identify the generating function $\sum _{n\ge 0}{b}_{n}{x}^{n}$

My try:

a) first. For the first element of each word there are $k$ possibilities. For every successor there are $$(k-1)$$ possibilities because they depend on the element before themselves.

Is this correct and complete?