Answer
$52488$
Work Step by Step
The provided sequence is $\frac{-8}{243},\frac{8}{81},\frac{-8}{27},\ldots $
A sequence is geometric if there exist a number $r$ called the common ratio for which,
$\frac{{{a}_{n+1}}}{{{a}_{n}}}=r$
Or ${{a}_{n+1}}={{a}_{n}}\cdot r$
${{a}_{n}}=a,ar,a{{r}^{2}},a{{r}^{3}},\ldots $.
The first term ${{a}_{1}}$ is ${{a}_{{}}}=\frac{-8}{243}$, and the second term ${{a}_{2}}$ is $a_2=\frac{8}{81}$,
Find the common ratio $r$, by using the formula $r=\frac{{{a}_{2}}}{{{a}_{1}}}$,
$\begin{align}
& r=\frac{\frac{8}{81}}{\frac{-8}{243}} \\
& =\frac{8}{81}\times \frac{243}{-8} \\
& =-3
\end{align}$
Apply the formula for the $n\text{th}$ term of a geometric sequence ${{a}_{n}}=a{{r}^{n-1}}$.
$\begin{align}
& {{a}_{14}}_{\text{th}}=\frac{-8}{243}{{\left( -3 \right)}^{14-1}} \\
& =\frac{-8}{243}{{\left( -3 \right)}^{13}}
\end{align}$
So, the value obtained is $52488$
Thus, the value of the $13\text{th}$ term of the geometric sequence is $52488$.
