Answer
The equation for the $n$th term is
$a_n=432(\frac{1}{6})^{n-1}$
and $a_7=\frac{1}{108}$
Work Step by Step
The given series is
$432,72,12,2,...$
First term $a_1=432$.
Common ratio $r=\frac{72}{432}=\frac{1}{6}$.
Equation for a geometric sequence is
$\Rightarrow a_n=a_1r^{n-1}$
Substitute $432$ for $a_1$ and $\frac{1}{6}$ for $r$.
$\Rightarrow a_n=432(\frac{1}{6})^{n-1}$
Substitute $7$ for $n$.
$\Rightarrow a_7=432(\frac{1}{6})^{7-1}$
Simplify.
$\Rightarrow a_7=432(\frac{1}{6})^{6}$
$\Rightarrow a_7=\frac{1}{108}$
