Answer
$a=-384$
The $n^{th}$ term of the given sequence is: $a_n = -384 \cdot \left(-\dfrac{3}{8}\right)^{n-1}$.
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:
$a_3=-54
\\a_6 =\dfrac{729}{256}$
Note that using the third term as reference point, the sixth term can be found by multiplying the common ratio $r$ three times to the value of the third term.
Thus,
$a_6=a_3 \cdot r \cdot r \cdot r
\\a_6 = a_3 \cdot r^3$
Since $a_3=-54$ and $a_6=\dfrac{729}{256}$, substituting these values gives:
$a_6=a_3 \cdot r^3
\\\dfrac{729}{256} = -54 \cdot r^3
\\\dfrac{1}{-54} \cdot \dfrac{729}{256} = \dfrac{1}{-54} \cdot (-54r^3)
\\-\dfrac{27}{512}= r^3
\\(-\frac{3}{8})^3 = r^3$
Take the cube root of both sides to obtain:
$-\dfrac{3}{8}=r$
With $r=-\dfrac{3}{8}$, the first term can be computed by dividing $a_3$ by the common ratio twice to obtain:
$a = a_3 \div (r \times r)
\\a = a_3 \div r^2
\\a=-54 \div (-\frac{3}{8})^2
\\a= -54 \div \frac{9}{64}
\\a = -54 \cdot \frac{64}{9}
\\a=-384$
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:
$a_n = -384 \cdot \left(-\dfrac{3}{8}\right)^{n-1}$
