Answer
A recursive rule for the sequence is
$a_1=243,a_n=\frac{1}{3}a_{n-1}$
Work Step by Step
The given sequence is
$243,81,27,9,3,...$
The first term is $a_1=243$.
Calculate ratio between each pair of consecutive terms.
$\frac{81}{243}=\frac{1}{3}$
$\frac{27}{81}=\frac{1}{3}$
$\frac{9}{27}=\frac{1}{3}$
$\frac{3}{9}=\frac{1}{3}$
The common ratio is $d=\frac{1}{3}$.
So, the sequence is geometric.
Recursive equation for a geometric sequence.
$a_n=r\cdot a_{n-1}$
Substitute $\frac{1}{3}$ for $r$.
$a_n=\frac{1}{3}a_{n-1}$
Hence, a recursive rule for the sequence is $a_1=243,a_n=\frac{1}{3}a_{n-1}$
