### Video Transcript

In this video weβre gonna look at
geometric sequences and weβre gonna see how to write down a general formula for a
particular geometric sequence. Then weβll go on and answer a few
typical questions. So letβs start off with the
definition. A sequence of numbers are called a
geometric sequence if each term is multiplied by the same common ratio to get the
next term. So for example, weβve got a
sequence of numbers three, six, 12, 24, and so on. And in this case, three is our
first term. And in each case, to get the next
number in the sequence, weβre simply doubling each term. So the common ratio is two. To get one term, we simply multiply
the previous term by two.

In another example, the sequence
10, 15, 22.5, 33.75, and so on, the first term in the sequence is 10. And I need to multiply each term in
the sequence by 1.5 to get the next term. So my common ratio is 1.5. Another example would be the
sequence seven, seven-tenths, seven hundredths, seven thousandths, and so on. Now for this sequence, my first
term will be seven. And I need to multiply each term by
a tenth to get the next term. So my common ratio is a tenth. And another example would be the
sequence 32, negative 16, eight, negative four. And in this case, the first term
will be 32. And Iβd need to multiply each term
by negative a half to get the next term. So my common ratio would be
negative a half. So my common ratio then could be
positive, could be negative, could be integer, fraction, decimal. Thereβs all sorts of possibilities
for what that might be.

Now depending on where you live,
there are several different styles of notation that are commonly used for geometric
sequences. For example, some people call the
first term π or π one or π’ one or π‘ in parentheses one, or π‘ one. But in this video, Iβm gonna use
the convention π one is the first term. And likewise, there are a different
ways that people express the πth term. It could be π π; it could be π’
π, π‘ π, or π‘ (π) written like this. Iβm gonna be using this one in this
video. But luckily, most people seem to
use π to represent the common ratio. So again thatβs what Iβll use.

So letβs write down the first five
terms of a geometric sequence with the first term, π one is equal to 12, and a
common ratio, π is equal to a third.

Well, weβre told that the first
term is 12. So letβs just write that down; the
first term is 12. Now, the common ratio is a third;
that means I have to multiply each term by a third in order to get the next
term. So to get the second term, I need
to multiply 12 by a third. And a third of 12 is four. And to get the third term, I need
to multiply the second term by a third. So thatβs four times a third which
is four over three, four-thirds. And to get the fourth term, Iβm
going to need to multiply that by a third. And four-thirds times a third is
four-ninths. So my fourth term is
four-ninths. And then letβs do that one more
time. Four-ninths times a third is four
twenty-sevenths. And that process carries on
forever. That geometric sequence carries on
for as many times as youβd like to write down.

Just going over the notation a bit
more, remember π one, the first term, is twelve; π two, the second term, is four;
π three, the third term, is four-thirds; π four, the fourth term, is four-ninths;
π five, the fifth term, is four twenty-sevenths; and so on. Now in order to get the second
term, we took the first term and multiplied it by the common ratio. And to get the third term, we took
the second term and multiplied it by the common ratio. So π three is π two times π. But remember, π two is π one
times π. So I can replace π two in here
with π one times π. So π two is π one times π. And then we multiply that by π to
get π three.

Now to get π four, we took π
three and multiplied that by the common ratio. So remember π three is π one
times π times π. So we multiplied that by π to get
π four. And likewise we multiply that by π
again to get the next term. So π five is π one times π times
π times π times π. Now, we can write that in power
format. So instead of writing π times π
times π times π, we can write π to the power of four. So π one is just twelve or π one,
π two is π one times π to the power of one, π three is π one times π to the
power of two, π four is π one times π to the power of three, and so on.

So Iβm just gonna complete that
sequence by saying π one is just π one times one. But instead of just writing one,
Iβm going to say π to the power of zero. Remember anything to the power of
zero is one. So now we got a bit of a
pattern. The first term is just the first
term times π to the power of zero. The second term is the first term
times π to the power of one. The third term is the first term
times π to the power of two. The fourth term is the first term
times π to the power of three. And the fifth term is the first
term times π to the power of four. So the power of π is one less than
the position of the term in the sequence.

So if I say π is the position in
the sequence, the πth term, π π, is simply the first term times π to the power
of one less than π. So thatβs π π equals π one times
π to the power of π minus one, one less than π. So now, weβve got a nice little
formula that tells us the any-any term in the sequence. So we donβt have to keep
multiplying by a third. We can go straight to that term in
the sequence.

So letβs see an example of
that.

So letβs use the general formula to
find the value of the seventh term in this particular sequence.

Well if the general term π π is
equal to π one times π to the power of π minus one. Thatβs the first term time the
common ratio- times the common ratio to the power of π minus one. Well, our first term was 12; we
were told in the question. So π one is 12. And our common ratio was a
third. So our general formula for this
particular sequence is π to the π is equal to 12 times a third to the power of π
minus one. So to find the seventh term, we put
π is equal to seven. And π seven, the seventh term, is
gonna be twelve times a third to the power of seven minus one. Well seven minus one is six. So the seventh term is going to be
12 times a third to the power of six. And a third to the power of six is
one over 729. So itβs gonna be 12 over one over
729, which is four over 243. Letβs have a look at another
question.

Write down the first term and the
common ratio for the following geometric sequence: 10, negative five, five over two,
negative five over four, and so on.

Well clearly, the first term is
equal to 10. So π one is equal to 10; that bit
was quite easy. And the common ratio is what do we
multiply each term by to get the next term. So Iβm just gonna label all of my
terms π one, π two, π three, and π four, and so on. And then Iβm just going to write a
little formula for how do I get from one term to the next term. Well, if I multiply the first term
by the common ratio, π, I get the second term. If I multiply the second term by
the common ratio, π, I get the third term. If I multiply the third term by the
common ratio, π, I get the fourth term, and so on and so on and so on. So looking at that first equation,
if I divide both sides of the equation by π one, I get that π is equal to π two
over π one. Now, if I divide both sides of the
second equation by π two, I get that π is equal to π three over π two and
similarly for the third equation.

So to work out the value of π, I
just take the value of one term and divide it by the value of the previous term. Now remember in a geometric
sequence, itβs a common ratio. So it doesnβt matter whether I take
the second and the first or the third and the second or the fourth and the
third. As long as I take two consecutive
terms, I will always find the same answer for π. Well looking at these numbers here,
the easiest pair to use is gonna be π one and π two. So π is equal to π two divided by
π one. π two is negative five and π one
is ten. So the common ratio is negative
five divided by 10 and that simplifies to negative a half.

So to get from each term to the
next term, I have to multiply by negative a half. 10 times negative a half is minus
five, negative five times negative a half is five over two, and so on. So these two facts here, π one
equals 10 and π equals negative a half, uniquely defines this sequence. When we know this, we can generate
all the terms in the sequence if weβre prepared to put in enough time and doing
enough multiplying.

Okay, letβs see some sequences and
try and work out whether or not they are geometric.

Well in fact, this question says,
is the following sequence arithmetic or geometric?

Now, if you remember, an arithmetic
sequence is one in which each term has a common difference added in order to
generate the next term. So to answer this question, we just
need to see what do we add to get from one term to the next and see if that is
constant and what do we multiply to get from one term to the next and see if that is
constant. So with this particular set of
numbers, if I add one every time, Iβm generating that sequence. But to multiply one term to get the
next term, the ratio keeps changing; itβs not a common ratio. So the fact that we got a common
difference means that this is an arithmetic sequence.

Letβs try this one then.

Is the following sequence
arithmetic or geometric? 11, 33, 99, 297, and so on.

Well, if we were adding we would
get- have to add different numbers each time to get the next number in the
sequence. But if we multiply each term by
three, we generate the next term. So weβve got a common ratio of
three. So the fact that weβve got a common
ratio rather than a common difference tells us that we got a geometric sequence.

Now this one, is the following
sequence arithmetic or geometric? One, two, four, seven, 11, and so
on. Well, I have to add a different
number each time to generate the next term. So itβs not an arithmetic
sequence. And I need to double the first to
get the second and double the second to get the third. But after that Iβm not doubling; I
am having to multiply by different numbers. So there is not a common difference
and there is not a common ratio. So itβs neither arithmetic or
geometric. So an interesting sequence, but
itβs not arithmetic or geometric.

Okay one last one of these
then.

Is the following sequence
arithmetic or geometric? 5.2, 5.2, 5.2, 5.2, and so on.

Well, what do you think? Is it arithmetic or is it
geometric? What do I have to add in order to
get from one term to the next? Well, itβs nothing in each
case. Iβm adding zero. So Iβve got a common difference of
zero. Well thatβs a bit weird. But that is an arithmetic
sequence. And what do I have to multiply each
term by to get the next term? Well in each case, Iβm just
multiplying by one. So again itβs a bit of a weird
sequence. But it is a geometric sequence
because we got a common ratio of one. So itβs a pretty weird example Iβll
grant you that. But it is both arithmetic and
geometric if we follow those strict rules. The common difference is zero and
the common ratio is one.

Now, letβs look at another
question.

Find the next three terms in the
geometric sequence 100, negative 10, one, negative 0.1, 0.01, and so on. So weβve got the first five terms,
π one, π two, π three, π four, and π five. And the first thing I need to do is
to work out what the common ratio is. What do I need to multiply π one
by to get π two and so on? And if you remember the way that we
do this is we divide one term by its previous term to find out what the ratio
is. And looking through there, I reckon
the second and third terms are gonna be the easiest ones to divide. And although you get the same
answer no matter which consecutive pair you divided, this oneβs easy because itβs
one divided by negative 10 which is negative a tenth.

So the common ratio is negative a
tenth. We need to multiply each term by
negative a tenth to get the next term. So to find the next three terms, I
just need to take the last term that we had and multiply that by negative a tenth,
then multiply that by negative a tenth, and multiply that by negative a tenth. So the sixth term is the fifth term
times negative a tenth; thatβs, 0.01 times negative a tenth, which is negative
0.01. So multiplying by negative a tenth
is the same because itβs dividing by negative 10. So this is relatively easy to
do. So the sixth term times negative a
tenth, the two negatives are gonna cancel out to make it positive. And 0.001 divided by 10 is
0.0001. And doing the same, the eighth term
is negative 0.00001. So we just need to write our answer
out there nice and clearly.

Now, we talked a bit about finding
a formula for a general term. So hereβs a question. So letβs have a look at this.

Find a formula for the general term
of the geometric sequence three, 15, 75, 375, 1875.

Well, our first term is three. So that bit is easy and Iβve got to
work out what the common ratio. And remember, weβre just gonna do a
division of one term divided by its previous term. And the easiest numbers to work
with here I think are gonna be these two, π one is three and π two is 15. So the common ratio is π two
divided by π one which is 15 over three which is five. Now remember, we were told in the
question that this is a geometric sequence. So it didnβt matter which pair of
terms β consecutive terms β that we chose; we would have got the same answer π
equals five. But just by choosing these first
two terms, the numbers were simpler.

So we know that π one, the first
term, is three and the common ratio is five. So we can put that into our
formula. And remember, to work out the value
of any particular term in the sequence, what we do is we take the first term and
weβre gonna keep multiplying it by the common ratio. Now, what we have to do if we are
looking for the fifth term, weβve only had to multiply that first term by the common
ratio four times in order to get that. So whatever term weβre looking for,
itβs the common ratio to the power of that term minus one. And we just worked out that π one
was three and π is five. So to work out the value of term π
in this particular sequence, itβs gonna be three times five to the power of whatever
term that is minus one, π minus one.

So letβs take that one little step
further now.

And weβve got to find a formula for
the general term of the geometric sequence negative 512, 128, negative 32, eight,
negative two, and so on. And we gotta use that formula to
find the value of the twelfth term of the sequence.

Or we can just read off the first
term there, negative 512. And now weβve gotta work out the
common ratio. So itβs any pair of consecutive
terms, one divided by the other, the second divided by the first. So Iβm gonna choose π four and π
five in this case cause they look like the easiest numbers to work with. So π five is negative two and π
four is eight. So the common ratio is gonna be
negative two over eight, which is negative a quarter. So now, Iβve got those two
important bits of information. Itβs pretty easy to work out the
general formula. So remembering π π is equal to π
one times π to the π minus one, letβs substitute the values π for π, π one and
π.

So my general formula is π π, the
nth term is equal to negative 512 times negative a quarter to the power of π minus
one. So Iβm trying to find now the
twelfth term. So π equals 12, which means that
the twelfth term, π 12, is equal to negative 512 times negative a quarter to the
power of 12 minus one. Well, 12 minus one is eleven. And when I work all that out, Iβve
got π 12 is equal to one over 8192.

So letβs quickly summarise what
weβve looked at there. The geometric sequence is where you
multiply each term by a common ratio to get the next term. For example, three, six, 12,
24. I double each term to get out the
next term. And in this case, the common ratio
is two and the first term was three. To work out the common ratio, which
weβve called π, you just take a term and divide it by the previous term. In general, the πth term is simply
the first term multiplied by π and minus one times. So in the case of our example, our
πth term will be three times two to the power of π minus one.