Answer
$a = \dfrac{81}{2}$
The $n^{th}$ term of the given sequence is: $a_n = \dfrac{81}{2} \cdot \left(\dfrac{2}{3}\right)^{n-1}$
Work Step by Step
To find the first and nth terms, the value of the common ratio $r$ is needed.
Note that the next term of a geometric sequence can be found by multiplying the common ratio $r$ to the current term.
The known terms are:
$a_4=12
\\a_7 =\dfrac{32}{9}$
Note that using the fourth term as reference point, the seventh term can be found by multiplying the common ratio $r$ three times to the value of the fourth term.
Thus,
$a_7=a_4 \cdot r \cdot r \cdot r
\\a_7 = a_4 \cdot r^3$
Since $a_4=12$ and $a_7=\dfrac{32}{9}$, substituting these values gives:
$a_7=a_4 \cdot r^3
\\\dfrac{32}{9} = 12 \cdot r^3
\\\dfrac{1}{12} \cdot \dfrac{32}{9} = \dfrac{1}{12} \cdot 12r^3
\\\dfrac{8}{27} = r^3
\\(\frac{2}{3})^3 = r^3$
Take the cube root of both sides to obtain:
$\dfrac{2}{3}=r$
With $r=\dfrac{2}{3}$, the first term can be computed by dividing $a_4$ by the common ratio three times to obtain:
$a = a_4 \div (r \times r \times r)
\\a = a_4 \div r^3
\\a=12 \div (\frac{2}{3})^3
\\a= 12 \div \dfrac{8}{27}
\\a = 12 \cdot \dfrac{27}{8}
\\a = \dfrac{81}{2}$
RECALL:
The $n^{th}$ term $a_n$ of a geometric sequence can be found using the formula
$a_n = a \cdot r^{n-1}$
where
$a$ = first term
$r$ = common ratio
Thus, the $n^{th}$ term of the given sequence is:
$a_n = \dfrac{81}{2} \cdot \left(\dfrac{2}{3}\right)^{n-1}$
