Answer
$a=648$
$a_7=\dfrac{1}{72}$
Work Step by Step
To find the first three terms, the value of the common ratio $r$ is needed.
Note that the previous term of a geometric sequence can be found by dividing the common ratio $r$ to the current term. This means that ti find the value of the first term, the third term must be divided by the common ratio twice (or by $r^2$).
The geometric sequence has:
$r=\frac{1}{6}$
$a_3=18$
To find the first term, divide $a_3$ by the common ratio $r$ twice (or by $r^2$) to obtain:
$a = \dfrac{a_3}{r^2}
\\a=\dfrac{18}{(\frac{1}{6})^2}
\\a=\dfrac{18}{\frac{1}{36}}
\\a=18 \cdot \dfrac{36}{1}
\\a= 648$
The $n^{th}$ term of a geometric sequence can be found using the formula $a_n=a \cdot r^{n-1}$ where $a$ = first term and $r$ = common ratio.
Thus, the $n^{th}$ term of the given sequence is given by the formula:
$a_n = a \cdot r^{n-1}
\\a_n = 648 \cdot \left(\dfrac{1}{6}\right)^{n-1}$
Thus, the 7th term of the sequence is:
$a_7 = 648 \cdot \left(\dfrac{1}{6}\right)^{7-1}
\\a_7 = 648 \cdot \left(\dfrac{1}{6}\right)^6
\\a_7=\dfrac{1}{72}$
