### Video Transcript

In this video, weβre gonna combine your knowledge of geometric sequences with
your knowledge of logarithms to solve some problems.

First of all, remember that a geometric sequence is a sequence of numbers in
which each term is multiplied by a common ratio, in order to obtain the next term. For example three, six, twelve,
twenty-four, and so on. The first term, which we generally call π one, in this case has
the value three. And in each case, Iβm multiplying one term by two to get the
next term. So that common multiple, we call the common ratio π, so in this case
π is equal to two. And we can come up with a formula for the πth term, π π is
equal to π one times π to the power of π minus one.

Now that works because our first term we called π one, and the second term was the first term times the common ratio, so thatβs
π one times π to the power of one. And to get our third term, we multiply that second term by π. So
weβve got π one times π times π, or times π squared. And to get our fourth term, we multiply that term by π again. So
weβve got π one times π squared times π, which is π one times π
cubed. So the first term is π one, not times π, or you
could say times one, which is π to the power of zero. So the first
term is π one times π to the power of zero, the second term is π one times
π to the power of one, the third term is π one times π to the power of
two, and the fourth term is π one times π to the power of three. So in
each case, itβs π one times π to the power of one less than that position in the
sequence, which is exactly what this formula is saying.

Now if youβre not told the value of π, you can easily work it
out. Because each term is simply π times the previous term, we can work out the
value of π by doing one term divided by the previous term. So the second term
divided by the first term will give you π, the third term divided by the second
term will also give you π, and the fourth term divided by the third term will
give you π. Each term divided by the previous term will always give you the same
answer, π.

And also remember, about logarithms, if π to the power of π₯ is equal to
π then log base π of π is equal to π₯. So log base π of π means, what power do I need to raise
π to in order to get π? Or, π to the power of what equals π? So I can use these two things combined to solve some geometric sequence
problems. Letβs take a look.

In the geometric sequence with first term seven and common ratio
two, which is the first term to exceed twelve thousand?

So our first term is seven, π one is equal to
seven. And the common ratio is two, so π is equal to two. Now remember, our general term for a geometric sequence is that the
πth term π π is equal to the first term π one times
the common ratio π to the power of π minus one. So in this case, it means that our πth term is equal to
seven times two to the power of π minus one. Now remember, this is a sequence so- and it really makes sense for
π to have integer values, the first term, the second term, the third term, and
so on. It doesnβt really make a lot of sense to talk about the twelve point six
twoth term for example.

Now we could try putting π π equal to twelve thousand, so the
πth term is twelve thousand, and see what value of π
we get. If itβs a whole number, then we found the term that equals twelve
thousand. So if we add one to that, the next term will be the first one
that exceeds it. And if π is not a whole number, then the next integer up from
that number will be the first term that exceeds twelve thousand. Letβs have a
look.

So this means that twelve thousand is equal to seven times two to the
power of π minus one. Well I could divide both sides by seven. And weβve got twelve thousand over seven is equal to two to the power
of π minus one. Well we could do a lot of trial and error with different values of
π to see which one matches this equation. But a better idea is to take
logarithms of both sides, and then we could use the rules of logarithms to help us solve the
equation. So Iβm gonna take logs base ten of both sides. So that log ten of twelve thousand over seven is equal to log ten of
two to the power of π minus one. Using the log power rule, log ten of two to the power of π minus
one is the same as π minus one times log ten of two. And now I could divide both sides by log ten of two, so that I
know what π minus one is equal to. And then I could add one to both sides so that Iβve just got an
equation for π.

And thatβs a nice little job for my calculator, so when I type that in β Now as we said before, it only makes sense for π to be an
integer. Weβre talking about positions in a sequence, so the first, second, third, and so on.
So if π was equal to eleven point seven, et cetera, et cetera, that
would generate an exact answer of twelve thousand. So really our only options in
terms of our sequence are, π could be eleven, or π
could be twelve. Well if π was eleven, then our term
would be lower than twelve thousand. So if π was
twelve, then the value would be higher than twelve thousand. So our
answer: The first term to exceed twelve thousand is the twelfth term.

So just quickly working out the value of the eleventh and the twelfth term,
you donβt actually need to do this by the way but weβre gonna do it just to show you how it
works, the eleventh term was seven thousand one hundred and sixty-eight and the
twelfth term was fourteen thousand three hundred and thirty-six. So as you can see, the eleventh term is less than twelve
thousand the twelfth term is the first term to be greater than twelve
thousand. So our answer is: The twelfth term fourteen thousand three hundred and
thirty-six is the first to exceed twelve thousand.

Letβs do one more example, this time one that involves depreciation. So itβs
a decreasing geometric sequence.

Ella buys a car for thirty thousand dollars. It depreciates, or
loses value, at a rate of six percent per year. How many years will it be before
itβs worth less than two thousand five hundred dollars?

Now this question is different to the last one in two important ways. Firstly
obviously, the numbers are going down rather than up because itβs depreciating geometric
sequence. But secondly, we have to be really careful about how we label the terms. Weβve been given a starting value of thirty thousand dollars,
but the first term in our sequence is how much it will be worth in one yearβs
time. Okay. So each year weβre losing six percent of the value of the
car, which means weβre retaining ninety-four percent of the value of the car. Now
to work out ninety-four percent of something, itβs ninety-four over a
hundred times that number. And ninety-four over a hundred is nought point nine
four. So itβs like if we multiply by nought point nine four, that will
tell us what the car will be worth in a yearβs time. So we could use a normal geometric sequence formula by saying that the first
term is thirty thousand times nought point nine four, and that common ratio is nought point nine four. And then the πth term of the sequence is π one times π to
the power of π minus one. And if we do that when π equals one, weβre finding the value in
one yearβs time. When π equals two, weβre finding value in
two yearsβ time. When π equals three, weβre finding value in
three yearsβ time, and so on.

But looking at that formula, weβre always multiplying thirty
thousand by nought point nine for once, and then another π minus
one times. So in total, weβre multiplying thirty thousand by
nought point four- nine four
π times. So to work out the residual value of the car in π yearsβ time is
just thirty thousand times nought point nine four to the power of π. Now before
we move on, Iβm just gonna look at another way to come to that same formula. In one yearβs time, the car will be worth nought point
nine four times thirty thousand. In two yearsβ time, weβll take the value after one
year and multiply that by nought point nine four, so itβs gonna be nought
point nine four times nought point nine four times thirty thousand, or nought
point nine four squared times thirty thousand. In three yearsβ time, weβre gonna take the value after
two years and multiply that by nought point nine four. So thatβs
nought point nine four times nought point nine four squared times thirty
thousand, which is nought point nine four cubed times thirty thousand. And in four years, weβll take the value after three
years and multiply that by nought point nine four again. So that will be
nought point nine four times nought point nine four cubed times thirty
thousand, or nought point nine four to the power of four times thirty
thousand.

Now looking at this first formula, nought point nine four,
thatβs the same as nought point nine four to the power of one. So what youβll
notice is that the power of nought point nine four is always equal to the year
number that weβre looking at. So nought point nine four cubed is the third year,
nought point nine four to the power of four is the fourth year, and so on. So the value after π years, letβs call it π π, is
equal to nought point nine four to the power of π times thirty thousand. And we want to know the value of π when this whole expression
becomes worth less than two thousand five hundred dollars. So thatβs the
calculation weβre gonna try to do. Well, except, weβre gonna do the same thing that we did last time. Weβre
actually gonna find- try and find the value of π that makes this equal to
two thousand five hundred, and then work out which is the integer value of π which takes
this value to below two thousand five hundred. So the car is worth two thousand five hundred dollars when
nought point nine four to the power of π is times thirty thousand is equal to two
thousand five hundred. Now if I divide both sides by thirty thousand, I get nought point nine four to the power of π is it equal to two
thousand five hundred over thirty thousand, and that cancels to
one-twelfth.

So nought point nine four to the power of π is equal to a
twelfth. Now Iβm gonna take log base ten of both sides. So log base ten of nought point nine four to the power of π is equal to
log base ten of a twelfth and now Iβm gonna use the log power rule to rearrange the
left-hand side. And that enables me to say that π times log ten of nought point nine
four is equal to log ten of a twelfth. So now I can divide both sides by log
ten of nought point nine four and that leaves me with an expression for
π. And again, I can just tap this into my calculator and then I find that
π would be equal to forty point one five nine four five four four and so on.

So this means that theoretically, if this was a continual thing, when
π is just over forty, so in just over forty yearsβ
time, the car will be worth exactly two thousand five hundred dollars. So that
means in forty yearsβ time, it will still be worth slightly more than two
thousand five hundred dollars. But in forty-one yearsβ time, the value
will have dipped to below two thousand five hundred dollars. So just quickly checking those two values, when π is forty, the
value of the car is two thousand five hundred and twenty-four dollars
eighty-five cents, just rounding to two decimal places obviously because itβs
money. And when π is forty-one, plugging that number into the equation, I get a
value of two thousand three hundred and seventy-three dollars and
thirty-six cents. So our answer is: The car will be worth less than two thousand five
hundred dollars in forty-one yearsβ time. Of course, it will also be
worth less than two thousand five hundred dollars in forty-two,
forty-three, and so on. So the first time it becomes worth less than two
thousand five hundred dollars, is in forty-one years.

So just a couple of tips and pointers then for things to watch out for, in
that sort of question. First, we were depreciating by six percent per year, now
thatβs the same as working out what is ninety-four percent of the number. So
rather than trying to take away six percent, we work out the residual value of
the ninety-four percent thatβs gonna remain. The other thing is, in these sorts of questions where youβre told an initial
value when something is happening every year, youβve gotta be very careful of what you mean
when you talk about π, whatβs your definition of the first term, whatβs your
definition of the value of π. The standard formula for geometric sequences is
π π equals π one times π to the power of π minus one. But because of our definition of π, and π, and
π one in this case, weβve got a formula that looks more like the
πth term of this sequence, is: Thirty thousand, initial value of
the car, times nought point nine four to the power of π. So you just need to be
careful about that discrepancy between π and π minus one.