tough inequality for a graph theory problem

tough inequality for a graph theory problem

I am stuck at a step in a graph theory problem. I have to prove that $$
\frac{d\cdot(d-1)^k}{(d-2)} \leq d^k$$ for $d,k \geq 3$. Here $d$ actually
refers to degree of a graph and $k$ the radius.
The inequality does not work for $d, k < 3$. how to prove this kind of
inequality for a particular range?