Video Transcript
True or False: The terms of a
geometric sequence can be plotted as a set of collinear points.
Let’s begin this question by
recalling that a geometric sequence is a sequence which has a common ratio between
any two consecutive terms. In order to consider whether a
geometric sequence might be collinear, which means lying on a straight line, then it
might be useful to take a few examples of geometric sequences.
So let’s take the sequence one,
three, nine, 27, and so on. We can say that it’s geometric
because the common ratio is three. Any term is found by multiplying
the previous term by three. If we were to graph these, then
we’d be graphing the 𝑛 or the index value alongside the term value. We could start with the coordinate
one, one because the first term has a value of one. The second coordinate would be that
of two, three. The term with index two has a value
of three.
However, a third coordinate of
three, nine might begin to reveal the pattern in this geometric sequence. We would not have a straight
line. In fact, what we would have would
be an exponential graph.
So let’s try another geometric
sequence. This time, let’s try a decreasing
geometric sequence. The sequence negative two, negative
four, negative eight, negative 16, and so on has a common ratio of two. Let’s try plotting these term
values. Once again, we can see that these
points would not lie on a straight line.
But there is another type of
geometric sequence, and that’s an alternating sequence. The signs of the terms alternate
between positive and negative. This is because the ratio is a
negative value. When we plot this geometric
sequence, we get a graph that looks like this. So far, none of the sequences that
we have considered have created a set of collinear points. So let’s consider what type of
sequence would.
Well, if we had a sequence which
produces a straight line, that means that when the index increases, the terms
increase by a constant or a constant difference. This type of sequence is actually
an arithmetic sequence. And it’s defined by a common
difference between any two consecutive terms. Remember that a geometric sequence
has a common ratio between terms, and that common ratio cannot be equal to one. It is therefore not possible that a
geometric sequence can be plotted as a set of colinear points. That means that the statement in
the question is false.
It’s worth noting that arithmetic
sequences are always linear but geometric sequences are never linear. In fact, they would create an
exponential function.
