To find a recursive rule for the sequence [tex]\(1, -6, 36, -216, \ldots\)[/tex], we need to observe the relationship between consecutive terms in the sequence.
First, let's calculate the ratios of consecutive terms:
1. The ratio between the second term [tex]\(-6\)[/tex] and the first term [tex]\(1\)[/tex]:
[tex]\[
\frac{-6}{1} = -6
\][/tex]
2. The ratio between the third term [tex]\(36\)[/tex] and the second term [tex]\(-6\)[/tex]:
[tex]\[
\frac{36}{-6} = -6
\][/tex]
3. The ratio between the fourth term [tex]\(-216\)[/tex] and the third term [tex]\(36\)[/tex]:
[tex]\[
\frac{-216}{36} = -6
\][/tex]
We see that each term is obtained by multiplying the previous term by [tex]\(-6\)[/tex].
Thus, the recursive rule for this sequence is:
[tex]\[
a_n = -6 \cdot a_{n-1}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{a_n=-6 \cdot a_{n-1}}
\][/tex]
