Which rule is a recursive rule for the sequence [tex]1, -6, 36, -216, \ldots?[/tex]

A. [tex]a_n = \frac{1}{6} \cdot a_{n-1}[/tex]

B. [tex]a_n = -\frac{1}{6} \cdot a_{n-1}[/tex]

C. [tex]a_n = 6 \cdot a_{n-1}[/tex]

D. [tex]a_n = -6 \cdot a_{n-1}[/tex]



Answer :

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]