To establish a recursive rule for the geometric sequence where the terms are [tex]\( 6, -24, 96, -384, \ldots \)[/tex]:
1. Identify the first term [tex]\( a_1 \)[/tex]:
[tex]\[
a_1 = 6
\][/tex]
2. Determine the common ratio [tex]\( r \)[/tex]:
The common ratio can be found by dividing the second term by the first term:
[tex]\[
r = \frac{-24}{6} = -4
\][/tex]
3. Write the recursive formula:
A geometric sequence can be defined recursively using the formula:
[tex]\[
a_n = a_{n-1} \times r
\][/tex]
Here, [tex]\( a_{n-1} \)[/tex] represents the previous term, and [tex]\( r \)[/tex] is the common ratio.
Putting this information together, the recursive rule for the given geometric sequence is:
[tex]\[
a_1 = 6
\][/tex]
[tex]\[
a_n = a_{n-1} \times -4
\][/tex]
