Alpha writes the infinite arithmetic sequence \[10, 8, 6, 4, 2, 0 ,\ldots.\] beta writes the infinite geometric sequence \[9, 6, 4, \frac{8}{3}, \frac{16}{9}, \ldots.\] gamma makes a sequence whose $n^{\text{th}}$ term is the product of the $n^{\text{th}}$ term of alpha's sequence and the $n^{\text{th}}$ term of beta's sequence: \[10\cdot 9 \quad,\quad 8\cdot 6\quad ,\quad 6\cdot 4\quad,\quad 4\cdot \frac83\quad,\quad 2\cdot \frac{16}{9}\quad,\quad \ldots.\] what is the sum of gamma's entire sequence?