\begin{align*} 7a_{n-1} - 10a_{n-2} = \amp 7(3\cdot 2^{n-1} + 7 \cdot 5^{n-1}) - 10(3\cdot 2^{n-2} + 7 \cdot 5^{n-2})\\ = \amp 21\cdot 2^{n-1} + 49 \cdot 5^{n-1} - 30\cdot 2^{n-2} - 70 \cdot 5^{n-2})\\ = \amp 21\cdot 2^{n-1} + 49 \cdot 5^{n-1} - 15\cdot 2^{n-1} - 14 \cdot 5^{n-1})\\ = \amp 6\cdot 2^{n-1} + 35 \cdot 5^{n-1}\\ = \amp 3\cdot 2^{n} + 7 \cdot 5^{n} = a_n \end{align*}