Video Transcript
Which of the following is not a
geometric sequence? Is it (A) 𝑤 over seven 𝑥,
negative one-sixth, seven 𝑥 over 36𝑤, negative 49𝑥 squared over 216𝑤 squared,
and so on? Is it (B) 11, negative 44, 176,
negative 704? (C) The sequence with terms log 𝑏,
log 𝑏 squared, log 𝑏 cubed, log 𝑏 to the fourth power. Or (D) the sequence with terms one
nineteenth, negative one over 57, one over 171, and negative one over 513.
Let’s begin by recalling what we
actually mean when we describe a sequence as geometric. A geometric sequence is one in
which each term is found by multiplying the previous term by some nonzero number,
and we call this a common ratio. For a sequence whose terms are 𝑎
sub 𝑛, the common ratio is found by dividing any term by the term that precedes it,
so 𝑎 sub 𝑛 plus one over 𝑎 sub 𝑛, or equivalently 𝑎 sub 𝑛 over 𝑎 sub 𝑛 minus
one. So what we can do is look at each
sequence and determine whether they have a common ratio.
Beginning with our first sequence,
we’re going to divide the second term by the first. That’s negative one-sixth divided
by 𝑤 over seven 𝑥. This is equivalent to negative
one-sixth times seven 𝑥 over 𝑤, which is negative seven 𝑥 over six 𝑤.
Let’s repeat this process with
terms two and three. The ratio of these terms is seven
𝑥 over 36𝑤 divided by negative one-sixth, which is equivalent to seven 𝑥 over
36𝑤 times negative six. This simplifies by dividing through
by that common factor of six to negative seven 𝑥 over six 𝑤. In fact, if we then also divide the
fourth term by the third, we get the exact same value, negative seven 𝑥 over six
𝑤. Since a common ratio exists between
consecutive terms in our first sequence, we know it must be geometric.
Let’s repeat this process for part
(B). Now, in fact, this requires very
little manipulation. Negative 44 divided by 11 is the
same as 176 divided by negative 44, which in turn is the same as negative 704
divided by 176. It’s negative four. So, once again, there exists a
common ratio between terms; this time, it’s negative four. And we can deduce part (B) is also
a geometric sequence.
Let’s move on to part (C). We’ll begin by dividing the second
term by the first. So that’s log 𝑏 squared divided by
log 𝑏. Using one of the laws of logs, we
can rewrite the numerator as two log 𝑏. Then, two log 𝑏 divided by log 𝑏
is simply equal to two. So the ratio of our first two terms
is two.
Let’s repeat this process with the
second and third term. That’s log 𝑏 cubed divided by log
𝑏 squared. We can rewrite this as three log 𝑏
divided by two log 𝑏. This time, three log 𝑏 divided by
two log 𝑏 is simply three over two. We’ve already seen that the ratio
of the first to second term and the second to third term are different. We don’t need to go ahead and check
the ratio of the third to fourth term. Since the common ratio does not
exist, we know that this cannot be a geometric sequence. So option (C) is not a geometric
sequence.
Let’s just double-check option
(D). The ratio of the first to second
terms is negative one over 57 divided by one nineteenth, and that’s negative
one-third. If we carry on in this way, we find
that one over 171 divided by negative one over 57 is also negative one-third. And the ratio of the third to
fourth term is also negative one-third. Since a common ratio exists, we’ve
proven that this is a geometric sequence as expected. So the answer is (C). It’s a sequence containing terms
log 𝑏, log 𝑏 squared, log 𝑏 cubed, and log 𝑏 to the fourth power.
