Let’s determine the seventh term of the given geometric sequence [tex]\(\sqrt{6}, -2\sqrt{3}, 2\sqrt{6}, -4\sqrt{3}, \ldots \)[/tex].
1. First, identify the first term ([tex]\(a\)[/tex]) of the sequence:
[tex]\[
a = \sqrt{6}
\][/tex]
2. To find the common ratio ([tex]\(r\)[/tex]), divide the second term by the first term:
[tex]\[
\text{second term} = -2\sqrt{3}
\][/tex]
[tex]\[
r = \frac{-2\sqrt{3}}{\sqrt{6}}
\][/tex]
3. Simplify the common ratio:
[tex]\[
r = \frac{-2\sqrt{3}}{\sqrt{6}} = \frac{-2\sqrt{3}}{\sqrt{3} \cdot \sqrt{2}} = \frac{-2\sqrt{3}}{\sqrt{3}\sqrt{2}} = \frac{-2}{\sqrt{2}} = -\sqrt{2}
\][/tex]
4. Now, to find the 7th term ([tex]\(a_7\)[/tex]) of the geometric sequence, use the formula for the [tex]\(n\)[/tex]-th term of a geometric sequence:
[tex]\[
a_n = a \cdot r^{n-1}
\][/tex]
With [tex]\(a = \sqrt{6}\)[/tex], [tex]\(r = -\sqrt{2}\)[/tex], and [tex]\(n = 7\)[/tex]:
[tex]\[
a_7 = \sqrt{6} \cdot (-\sqrt{2})^{7-1}
\][/tex]
5. Simplify the exponent:
[tex]\[
a_7 = \sqrt{6} \cdot (-\sqrt{2})^6
\][/tex]
6. Since [tex]\((- \sqrt{2})^6 = (\sqrt{2})^6\)[/tex] (an even power removes the negative sign) and [tex]\((\sqrt{2})^6 = (2^{1/2})^6 = 2^{3} = 8\)[/tex]:
[tex]\[
a_7 = \sqrt{6} \cdot 8 = 8 \cdot \sqrt{6}
\][/tex]
Therefore, the seventh term of the sequence is:
[tex]\[
8\sqrt{6}
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{8\sqrt{6}}
\][/tex]
