Answer :
1) Given this Explicit formula, let's find a_1, and then a_2 and then compare both terms:
[tex]\begin{gathered} a_n=2(-2x^2)^n \\ a_1=2(-2x^2)^1\Rightarrow a_1=-4x^2 \\ a_2=2(-2x^2)^2\Rightarrow a_2=2(4x^4)=8x^4 \\ a_3=2(-2x^2)^3\Rightarrow a_3=2(-8x^6)=-16x^6 \end{gathered}[/tex]2) Comparing both terms we can state that this is a Geometric Sequence we can write its ratio as:
[tex]q=\frac{8x^4}{-4x^2}=-2x^2[/tex]3) So we can write our Recursive formula as:
[tex]\begin{gathered} a_n=-2x^2\times a_{n-1} \\ \text{Testing:} \\ a_2=-2x^2\times a_1 \\ a_2=-2x^2\times-4x^2 \\ a_{2=\text{ }}8x^4 \end{gathered}[/tex]So the answers are:
[tex]\begin{gathered} a_1=-4x^2 \\ a_n=-2x^2\times a_{n-1} \end{gathered}[/tex]