Question

A5=2500 and a8=312,500 in explicit form geometric Sequence and series

Answer 1

If $a_n$ is a geometric sequence, then there is some $r$ for which

$a_n=ra_{n-1}$

We have

$a_8=ra_7=r^2a_6=r^3a_5$

so

$312,500=2500r^3\implies r^3=125\implies r=5$

Then

$a_5=5a_4=5^2a_3=5^3a_2=5^4a_1\implies a_1=4$

So this sequence is recursively defined by

$\begin{cases}a_1=4\\a_n=5a_{n-1}&\text{for }n>1\end{cases}$

and explicitly by

$a_n=4\cdot5^{n-1}$