Given the sequence below:
[tex]5,-25,125,\ldots[/tex]
It can be observed that the sequence shown is a geometric sequence.
The nth term of a geometrical sequence is given by the formula below:
[tex]a_n=ar^{n-1}[/tex][tex]\begin{gathered} \text{where } \\ a_n=\text{nth term} \\ a=\text{first term} \\ r=\text{common ratio} \end{gathered}[/tex]
From the given sequence, we can determine a, and r as shown below
[tex]\begin{gathered} a=5 \\ r=\frac{a_2}{a_1}=\frac{-25}{5}=-5 \end{gathered}[/tex]
Substitute for a and r in the formula to get the nth term as shown below
[tex]a_n=5(-5)^{n-1}[/tex][tex]a_n=5(-\frac{1}{5})^{1-n}[/tex]
Hence, the nth term is, as shown above
