Let's break down the problem and provide a detailed solution for each part.
### Part (a)
We are given the first three terms of a sequence:
[tex]\[ 2x, \ y^2, \ 2xy^2 \][/tex]
The sequence rule states that each term is derived by multiplying the two previous terms.
- To find the 4th term, we multiply the 2nd and 3rd terms:
[tex]\[ \text{4th term} = y^2 \cdot 2xy^2 = 2x \cdot y^4 = 2xy^4 \][/tex]
- To find the 5th term, we multiply the 3rd and 4th terms:
[tex]\[ \text{5th term} = 2xy^2 \cdot 2xy^4 = 4x^2y^6 \][/tex]
- To find the 6th term, we multiply the 4th and 5th terms:
[tex]\[ \text{6th term} = 2xy^4 \cdot 4x^2y^6 = 8x^3y^{10} \][/tex]
Thus, the 6th term of the sequence is:
[tex]\[ 8x^3y^{10} \][/tex]
Therefore, the correct choice is:
[tex]\[ \boxed{C} \][/tex]
### Part (b)
We are given that the 7th term is [tex]\(32x^5y^{16}\)[/tex] and this term is negative.
Since the product is negative, either [tex]\(x\)[/tex] or [tex]\(y\)[/tex] must be negative, but not both. This means one of them must be positive, and the other must be negative.
- [tex]\(x\)[/tex] can be either positive or negative.
- [tex]\(y\)[/tex] can be either positive or negative, as one or the other needs to be negative to result in a negative product for odd terms while having positive powers.
Thus, for:
```
\begin{tabular}{|c|c|c|c|}
\cline { 2 - 4 } \multicolumn{1}{c|}{} & \begin{tabular}{c}
Must be \\
positive
\end{tabular} & \begin{tabular}{c}
Must be \\
negative
\end{tabular} & Either \\
\hline [tex]$x$[/tex] & & & x \\
\hline [tex]$y$[/tex] & & & x \\
\hline
\end{tabular}
```
