$$\begin{gathered} q^5=1024 \\ q=4 \\ \{\begin{array}{l}q=4\\ a_1\cdot q^2=\frac{3}{2}\end{array} \\ {a}_1\cdot 4^2=\frac{3}{2} \\ {a}_1\cdot 16=\frac{3}{2}|:16 \\ {a}_1=\frac{3}{32} \\ \{\begin{array}{l}q=4\\ a_1=\frac{3}{32}\end{array} \end{gathered}$$

$a_n=\frac{3}{32}\cdot 4^{n-1}=\frac{3}{2^5}\cdot {\left(2^2\right)}^{n-1}=\frac{3}{2^5}\cdot 2^{2(n-1)}=\frac{3\cdot 2^{2n-2}}{2^5}=3\cdot 2^{2n-2-5}=3\cdot 2^{2n-7}$