Answer
$\dfrac{1}{3^{n-2}}$
Work Step by Step
We are given the geometric sequence:
$a_3=\dfrac{1}{3}$
$a_6=\dfrac{1}{81}$
Determine the common ratio $r$:
$\dfrac{a_3}{a_6}=\dfrac{\dfrac{1}{3}}{\dfrac{1}{81}}$
$\dfrac{a_1r^2}{a_1r^5}=27$
$r^3=\dfrac{1}{27}$
$r=\dfrac{1}{3}$
Determine $a_1$:
$a_3=a_1r^2$
$\dfrac{1}{3}=a_1\cdot \left(\dfrac{1}{3}\right)^2$
$a_1=\dfrac{\dfrac{1}{3}}{\dfrac{1}{9}}$
$a_1=3$
Determine $a_n$:
$a_n=a_1r^{n-1}=3\left(\dfrac{1}{3}\right)^{n-1}=\dfrac{1}{3^{n-2}}$
