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)`