Answer
$2734375$
Work Step by Step
The provided sequence is $\frac{7}{625},\frac{-7}{125},\frac{7}{25},\ldots $,
Thus,
$\frac{{{a}_{n+1}}}{{{a}_{n}}}=r$
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 $a=\frac{7}{625}$, and the second term ${{a}_{2}}$ is $a_2=\frac{-7}{125}$ ,
Find the common ratio $r$, by using the formula $r=\frac{{{a}_{2}}}{{{a}_{1}}}$
$\begin{align}
& r=\frac{\frac{-7}{125}}{\frac{7}{625}} \\
& =\frac{-7}{125}\times \frac{625}{7} \\
& =-5
\end{align}$
Substitute the value of the first term and the common ratio in the above equation,
$\begin{align}
& {{a}_{13}}_{\text{th}}=\frac{7}{625}{{\left( -5 \right)}^{13-1}} \\
& =\frac{7}{625}{{\left( -5 \right)}^{12}}
\end{align}$
To find the value of $\frac{7}{625}{{\left( -5 \right)}^{12}}$, use a calculator
So, the value obtained is $2734375$
Thus, the value of the $13\text{th}$ term of the geometric sequence is $2734375$.
