Question

How do you find nth term rule for `a_2=5` and `a_4=1/5`?

Answer 1

There are two possibilities:

`a_n = 25(1/5)^(n-1)`

`a_n = -25(-1/5)^(n-1)`

Explanation:

Assuming that this is a geomtric sequence, since that's the topic under which it is posted...

The general formula for the `n`th term of a geometric sequence is:

`a_n = ar^(n-1)`

where `a` is the initial term and `r` the common ratio.

In our example, we find:

`r^2 = (ar^3)/(ar^1) = a_4/a_2 = (1/5)/5 = 1/25`

Hence `r = +-sqrt(1/25) = +-1/5`

If `r=1/5` then `a = (ar)/r = a_2/r = 5/(1/5) = 25`

If `r = -1/5` then `a = (ar)/r = a_2/r = 5/(-1/5) = -25`