### Video Transcript

Find the geometric sequence and the
sum of the first five terms given the sum of the first three terms is one and the
sum of the next three terms is 27.

In this question, weβre asked to
find two things. First, weβre asked to find a
geometric sequence, and weβre also asked to find the sum of the first five terms of
this geometric sequence. To do this, weβre given two pieces
of information. Weβre told the sum of the first
three terms of this geometric sequence is one, and weβre also told the sum of the
next three terms of this geometric sequence is 27.

To answer this question, we start
by recalling a geometric sequence is a sequence where the ratio between successive
terms is constant. And this means to generate terms in
our sequence, we multiply by a constant ratio π. For example, if the first term of
our sequence is π, the second term will be π times π and the third term will be
ππ multiplied by π, which is ππ squared. And this pattern continues. The first six terms of a geometric
sequence with initial term π and ratio of successive terms π is given as
follows.

Therefore, to answer the first part
of this question of determining the geometric sequence, we need to determine the
value of π, the first term of this sequence, and the ratio of successive terms
π. To do this, we need to use the two
pieces of information weβre given. First, weβre told the sum of the
first three terms is one. And thereβs a few different ways of
doing this. We could just add the first three
terms of this geometric sequence together. However, weβll do this by using a
fact about summing finite geometric sequences.

The sum of the first π terms of a
geometric sequence with first term π and ratio of successive terms π, π sub π,
is equal to π multiplied by one minus π to the πth power all divided by one minus
π provided π is not equal to one. And to apply this in our question,
we do need to check if the ratio is equal to one. However, we can see that this is
not true. If the ratio π is just equal to
one, every single term in the geometric sequence is equal to π. And in this case, the first three
terms of the geometric sequence would all be π and the next three terms of the
geometric sequence would also be π. And therefore, the sum of the first
three terms would be equal to the sum of the next three terms. This is not the case in this
question, so π is not equal to one.

Therefore, we can apply this
formula to the sequence. Since the sum of the first three
terms of our sequence is equal to one, one must be equal to π multiplied by one
minus π cubed all divided by one minus π. Weβre also told the sum of the next
three terms of this geometric sequence is 27. And thereβs a few different ways of
using this. We could just add the three terms
of this sequence together and set this equal to 27. We could also use the fact that the
sum of these three terms will be the sum of the first six terms of this sequence,
where we remove the sum of the first three terms of this sequence. And both of these methods would
work, and we can use them if we prefer. However, weβre going to use a
different method.

We can see that these three terms
in this geometric sequence form their own geometric sequence. The ratio of successive terms is π
and the first term is ππ cubed. And π is not equal to one, so we
can add them together using our formula. The first term is ππ cubed. And weβre adding three terms of a
geometric sequence together, and we know this is equal to 27. This gives us 27 is equal to ππ
cubed multiplied by one minus π cubed all divided by one minus π. We now have two equations in two
unknowns. We need to solve this to find the
values of π and π.

And thereβs many different ways of
solving these equations. Weβll only go through one of
these. Weβre going to multiply the top
equation through by 27. On the left-hand side, weβll get
27, and on the right-hand side, weβll get an extra factor of 27. This gives us the following
equation. Now, the left-hand side of both of
our equations are equal. So the right-hand sides of these
two equations must also be equal. And if we equate the right-hand
side of both of these two equations, we get the following equation. And we can now solve this for the
value of π.

First, remember when we considered
our sequence, we were able to show that the value of π was not equal to one. Therefore, we can multiply both
sides of this equation through by one minus π. We can also divide both sides of
the equation through by one minus π cubed. Next, we want to divide both sides
of our equation through by π. However, to do this, we need to
check if π is equal to zero. And in fact we can show that π is
nonzero in exactly the same way we did to show that π was not equal to one. If π was zero, every single term
in our geometric sequence would be equal to zero. And in this case, the sum of the
first three terms will be zero. Therefore, π cannot be equal to
zero. And if π is not equal to zero, we
can divide both sides of our equation through by π, giving us that 27 is equal to
π cubed.

Finally, we take the cube root of
both sides of this equation to see that π is equal to three. We still need to find the value of
π. We can do this by substituting π
is equal to three into either of our original equations. For example, we can substitute π
is equal to three into the top equation. We get 27 is equal to 27π
multiplied by one minus three cubed all divided by one minus three. And if we evaluate and rearrange
this equation for π, we would find that π is equal to one divided by 13.

Now that weβve determined the
initial term of the sequence and the ratio of successive terms, we have enough to
determine all of the terms of the geometric sequence. The first term π is one over 13
and the ratio of successive terms π is equal to three. So we multiply one over 13 by
factors of three to generate the terms of the sequence. π sub π is one over 13, three
over 13, nine over 13, and the sequence continues.

The second part of this question
wants us to determine the sum of the first five terms of this sequence. And thereβs a few different ways we
could do this. We could just find the first five
terms of the sequence and add them together, or we could just use our formula. Substituting π is equal to one
over 13, π is equal to three, and π is equal to five into this formula, we get π
sub five is equal to one over 13 multiplied by one minus three to the fifth power
all divided by one minus three, which if we evaluate weβll see is equal to 121
divided by 13, which gives us our final answer.

If we have a geometric sequence
where the sum of the first three terms is one and the sum of the next three terms is
27, then we were able to show the geometric sequence π sub π would be one over 13,
three over 13, nine over 13, and the sequence continues. And the sum of the first five terms
π sub five would be 121 divided by 13.