Question

How do you find r and a1 for the geometric sequence: a3 = 5, a8 = 1/625?

Answer 1

`r=1/5`
`a_1=125`

Explanation:

For a geometric series:
`color(white)("XXX")a_m=a_n*r^(n-m)`

We will use this general formula in two forms
`color(white)("XXX")a_8=a_3*r^5`
and
`color(white)("XXX")a_1=a_3*r^(-2)`

We are told `a_8 = 1/625` and `a_3=5`
Therefore
`color(white)("XXX")1/625 = 5*r^5`

`color(white)("XXX")r^5=1/(5*625) = 1/(5*5*4) = 1/(5^5)`

`color(white)("XXX")r=1/5`

and since `a_1=a_3*r^(-2)`
`color(white)("XXX")a_1= 5*(1/5)^(-2) = 5*5^2=125`