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?

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, ...#