Question

How do you determine whether the sequence `1, -1/2, 1/4, -1/8,...` is geometric and if it is, what is the common ratio?

Answer 1

The first four terms are in geometric progression with common ratio `-1/2`. So what we are given is consistent with the possibility that this is a geometric sequence.

Explanation:

Given:

`1, -1/2, 1/4, -1/8,...`

Note that:

`(-1/2)/1 = -1/2`

`(1/4)/(-1/2) = -1/2`

`(-1/8)/(1/4) = -1/2`

So there is a common ratio `-1/2` between each successive pair of terms.

That qualifies:

`1, -1/2, 1/4, -1/8`

as a geometric sequence.

We can write the formula for the general term of this sequence as:

`a_n = (-1/2)^(n-1)`

If the `...` does continue in the same way then the whole infinite sequence indicated is a geometric sequence.

Note however that this is a slight presumption. The fact that the first four terms are in geometric progression does not force the rest to be.

We might argue that `...` means just that: that the sequence carries on in a similar way, but that results in a slightly circular argument. It presupposes that we have identified the intended pattern.

For example, consider the sequence:

`1/3, 2/9, 3/27,...`

What does the `...` signify?

One possibility is that the standard term is `n/3^n`, giving us:

`1/3, 2/9, 3/27, 4/81, 5/243, 6/729,...,n/3^n,...`

Another is that this is simply an arithmetic sequence with initial term `1/3=3/9` and common difference `-1/9`, writable as:

`3/9, 2/9, 1/9, 0/9, -1/9, -2/9,..., (4-n)/9, ...`