To solve the problem, let's look at the given sequence and the rule for generating the subsequent terms.
The initial terms given are:
1. [tex]\(2x\)[/tex]
2. [tex]\(y^2\)[/tex]
3. [tex]\(2xy^2\)[/tex]
The rule for generating the sequence is to multiply the previous two terms to obtain the next term.
Let's calculate the next terms step by step.
Fourth Term:
To find the fourth term, we multiply the second term ([tex]\(y^2\)[/tex]) by the third term ([tex]\(2xy^2\)[/tex]):
[tex]\[
(y^2) \times (2xy^2) = 2xy^4
\][/tex]
Fifth Term:
To find the fifth term, we multiply the third term ([tex]\(2xy^2\)[/tex]) by the fourth term ([tex]\(2xy^4\)[/tex]):
[tex]\[
(2xy^2) \times (2xy^4) = 4x^2y^6
\][/tex]
Sixth Term:
To find the sixth term, we multiply the fourth term ([tex]\(2xy^4\)[/tex]) by the fifth term ([tex]\(4x^2y^6\)[/tex]):
[tex]\[
(2xy^4) \times (4x^2y^6) = 8x^3y^{10}
\][/tex]
Therefore, the sixth term of the sequence is:
[tex]\[
8x^3y^{10}
\][/tex]
Finally, we can select the correct option:
[tex]\[
\boxed{C \ 8 x^3 y^{10}}
\][/tex]
