# 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?

Aug 23, 2017

The first four terms are in geometric progression with common ratio $- \frac{1}{2}$. So what we are given is consistent with the possibility that this is a geometric sequence.

#### Explanation:

Given:

$1 , - \frac{1}{2} , \frac{1}{4} , - \frac{1}{8} , \ldots$

Note that:

$\frac{- \frac{1}{2}}{1} = - \frac{1}{2}$

$\frac{\frac{1}{4}}{- \frac{1}{2}} = - \frac{1}{2}$

$\frac{- \frac{1}{8}}{\frac{1}{4}} = - \frac{1}{2}$

So there is a common ratio $- \frac{1}{2}$ between each successive pair of terms.

That qualifies:

$1 , - \frac{1}{2} , \frac{1}{4} , - \frac{1}{8}$

as a geometric sequence.

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

${a}_{n} = {\left(- \frac{1}{2}\right)}^{n - 1}$

If the $\ldots$ 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 $\ldots$ 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:

$\frac{1}{3} , \frac{2}{9} , \frac{3}{27} , \ldots$

What does the $\ldots$ signify?

One possibility is that the standard term is $\frac{n}{3} ^ n$, giving us:

$\frac{1}{3} , \frac{2}{9} , \frac{3}{27} , \frac{4}{81} , \frac{5}{243} , \frac{6}{729} , \ldots , \frac{n}{3} ^ n , \ldots$

Another is that this is simply an arithmetic sequence with initial term $\frac{1}{3} = \frac{3}{9}$ and common difference $- \frac{1}{9}$, writable as:

$\frac{3}{9} , \frac{2}{9} , \frac{1}{9} , \frac{0}{9} , - \frac{1}{9} , - \frac{2}{9} , \ldots , \frac{4 - n}{9} , \ldots$