A note on sum of k-th power of Horadam's sequence

Let $w_{n+2}=p{w}_{n+1}+q{w}_n$ for $n\ge 0$ with $w_0=a$ and $w_1=b$. In this paper we find an explicit expression, in terms of determinants, for ${\sum }_{n\ge 0}{w}_n^k{x}^n$ for any $k\ge 1$. As a consequence, we derive all the previously known results for this kind of problems, as well as many new results.