How to solve the following relationship?

Question

We can easily seen that $${q^{2n}} = {q^n}\left( {{q^n} - 1} \right) + {q^n},{q^{3n}} = {q^n}{\left( {{q^n} - 1} \right)^2} + 2{q^n}\left( {{q^n} - 1} \right) + {q^n}.$$ Similarly, we assume that $${q^{mn}} = {q^n}\sum\limits_{k = 1}^{m - 1} {{a_k}{{\left( {{q^n} - 1} \right)}^k}} + {q^n}.$$ then, how to obtain the coefficient $a_k$.