Answer
$a_{1}=648,\displaystyle \quad a_{7}=\frac{1}{72}$
Work Step by Step
A geometric sequence is a sequence whose terms are obtained by multiplying each term by the same fixed constant $r$ to get the next term.
A geometric sequence has the form
$a, ar, ar^{2}, ar^{3}, \ldots$
The number $a$ is the first term of the sequence, and the number $r $is the common ratio.
The nth term of the sequence is $\quad a_{n}=ar^{n-1}$
--------------
Given: $r=\displaystyle \frac{1}{6},\quad a_{3}=18,$
Using $a_{n+1}=a_{n}r$
$a_{3}=a_{2}\displaystyle \cdot\frac{1}{6}$ leads to
$a_{2}=18\times 6=108$
$a_{2}=a_{1}r$ leads to
$a_{1}=a=108\times 6=648$
$(648=3\times 216=3\cdot 6^{3})$
So, $a_{n}=ar^{n-1}=648(\displaystyle \frac{1}{6})^{n-1}$
$a_{7}=648(\displaystyle \frac{1}{6})^{6}=\frac{3\cdot 6^{3}}{6^{6}}=\frac{3}{216}=\frac{1}{72}$
