Question

What is the 7th term of the geometric sequence where a1 = –4 and a5 = –1,024?

Answer 1

`-16384` or `16384`

Explanation:

The common ratio `r` of this geometric sequence is a `4`th root of `(-1024)/(-4) = 256 = 4^4`, since `a_5 = r^4 a_1`

The possible Real `4`th roots are `+-4`

In either case:

`a_7 = r^2 a_5 = 16*(-1024) = -16384`

So why do I say that `a_7` may be `16384`?

The Complex numbers `+-4i` are also `4`th roots of `256`.

If `r = +-4i` then `r^2 = -16` and we find:

`a_7 = r^2 a_5 = (-16)*(-1024) = 16384`