If [tex]a_n[/tex] is a geometric sequence, then there is some [tex]r[/tex] for which
[tex]a_n=ra_{n-1}[/tex]
We have
[tex]a_8=ra_7=r^2a_6=r^3a_5[/tex]
so
[tex]312,500=2500r^3\implies r^3=125\implies r=5[/tex]
Then
[tex]a_5=5a_4=5^2a_3=5^3a_2=5^4a_1\implies a_1=4[/tex]
So this sequence is recursively defined by
[tex]\begin{cases}a_1=4\\a_n=5a_{n-1}&\text{for }n>1\end{cases}[/tex]
and explicitly by
[tex]a_n=4\cdot5^{n-1}[/tex]
