Answer
A recursive rule for the sequence is $a_1=128,a_n=-\frac{1}{4}a_{n-1}$
Work Step by Step
The given sequence is
$128,-32,8,-2,0.5,...$
The first term is $a_1=128$.
Calculate ratio between each pair of consecutive terms.
$\frac{-32}{128}=-\frac{1}{4}$
$\frac{8}{-32}=-\frac{1}{4}$
$\frac{-2}{8}=-\frac{1}{4}$
$\frac{0.5}{-2}=-\frac{1}{4}$
The common ratio is $r=-\frac{1}{4}$.
So, the sequence is geometric.
Recursive equation for a geometric sequence.
$a_n=r\cdot a_{n-1}$
Substitute $-\frac{1}{4}$ for $r$.
$a_n=-\frac{1}{4}a_{n-1}$
Hence, a recursive rule for the sequence is $a_1=128,a_n=-\frac{1}{4}a_{n-1}$
