Video Transcript
Find the fifth term of the
geometric sequence one over 86, negative one over 43, two over 43, and so on.
We know that this is a geometric
sequence. But we haven’t been given any rule
to find each term in this sequence. So we’re going to need to look for
a pattern, a rule which gets us from one term to the next. As this is a geometric sequence, we
know that there’s a certain value that we multiply each term by to get the next
term. And we call this the common ratio
𝑟.
So let’s try to find what this
value is. We could do this just by
observation. Or another way we could do this is
to take a term and divide it by the previous term. So, if we take the second term,
which is negative one over 43, and divide it by the first term, which is one over
86, we get the value for 𝑟. We know that we can divide two
fractions by changing the sign from a divide to a times and flipping the second
fraction. So this is the same as negative one
over 43 multiplied by 86 over one. That gives us negative 86 over 43,
but 86 divided by 43 is just two. So we found the value of 𝑟 to be
negative two.
And by looking at the second term
and the third term, we can see that multiplying by negative two also works for these
two terms. So we know we found the right value
for 𝑟. So let’s go ahead and find the
fourth term and then the fifth term. Two over 43 multiplied by negative
two just gives us negative four over 43. And be careful, we’re not quite
done with the question. We’re looking for the fifth term of
this sequence. So taking negative four over 43 and
multiplying it by negative two gives us eight over 43. And in fact the sequence doesn’t
stop there. We don’t know what the rest of the
terms are. But the sequence will just continue
from here with the same pattern. But our final answer is the fifth
term of this sequence, which we’ve found to be eight over 43.
