Video Transcript
Applications of Geometric Sequences
and Series
In this lesson, we’re going to see
a lot of real-world problems which involve geometric sequences and series. We’re going to apply everything we
know about these to try and solve these real-world problems. We’ll see how to find the common
ratio, how to find an explicit formula for the 𝑛th term, the order and value of the
specific terms of the sequences, and how to find the sum of any number of given
terms of our sequence.
Before we start tackling real-world
problems, let’s start by recalling everything we know about geometric sequences and
series. First, we know that geometric
sequences start with an initial value and we usually denote this 𝑎. Next, in a geometric sequence, we
know there’s always a common ratio between any two successive terms, and we usually
call this ratio 𝑟. And equivalent to this, we can also
say we multiply the previous term by 𝑟 to get the next term in our sequence.
And on top of this, we could also
have some formula to help us find information about geometric sequences and
series. If we let 𝑢 sub 𝑛 be the 𝑛th
term in a geometric sequence with initial value 𝑎 and ratio of successive terms 𝑟,
then we know 𝑢 sub 𝑛 is equal to 𝑎 times 𝑟 to the power of 𝑛 minus one. Because we just need to multiply
our initial value by 𝑟 𝑛 minus one times to get the 𝑛th term in our sequence.
Next, we also know we can find the
ratio of successive terms in a geometric sequence by taking the quotient of two
successive terms. 𝑟 will be equal to 𝑢 sub 𝑛 plus
one divided by 𝑢 sub 𝑛, and of course, this is provided that 𝑛 is greater than or
equal to one. We’ve also seen how to take the sum
of the first 𝑛 terms of a geometric sequence. We denote this 𝑆 sub 𝑛. We’ve shown that the sum of the
first 𝑛 terms of this geometric sequence will be 𝑎 times one minus 𝑟 to the 𝑛th
of power divided by one minus 𝑟. And there is a few things worth
pointing out about this formula. First, it’s not valid when our
value of 𝑟 is equal to one, because then we’re dividing by zero. But if 𝑟 was equal to one, our
geometric sequence would just be 𝑎 over and over again. So, normally, we wouldn’t think
about this as a geometric sequence anyway.
Next, there is an equivalent
formula we get by multiplying the numerator and denominator through by negative one,
which is 𝑎 multiplied by 𝑟 to the 𝑛th power minus one all over 𝑟 minus one. And both of these will give the
correct answer for any value of 𝑟 not equal to one. However, usually, we use the first
version when our value of 𝑟 is less than one and the second version when 𝑟 is
greater than one. However, it is personal preference
which one you want to use.
And finally, we’ve also seen how to
add an infinite number of terms of our geometric sequence together. We usually denote this 𝑆 sub
∞. And we know it’s equal to 𝑎
divided by one minus 𝑟 if the absolute value of our ratio 𝑟 is less than one. And if the absolute value of 𝑟 is
greater than or equal to one, our series will be divergent. Although there is one tiny caveat
with this. We do need to check our value of 𝑎
is nonzero because then otherwise our sequence would just be zero over and over
again. However, usually, we wouldn’t think
of this as a geometric sequence anyway.
Let’s now see some examples of how
we can apply all of this to find information about real-world problems.
Chloe joined a company with a
starting salary of 28,000 dollars. She receives a 2.5 percent salary
increase after each full year in the job. The total Chloe earns over 𝑛 years
is a geometric series. What is the common ratio? Write a formula for 𝑆 sub 𝑛, the
total amount in dollars Chloe earns in 𝑛 years at the company. After 20 years with the company,
Chloe leaves. Use your formula to calculate the
total amount she earned there.
And there’s one more part to this
question we will discuss later. In this question, we’re given a
real-world problem involving the amount of money Chloe will earn at a company over a
period of time. We’re told that Chloe has a
starting salary of 28,000 dollars and that she receives a 2.5 percent salary
increase after every full year she is in her job. In fact, this is enough information
to determine that the amount she earns in 𝑛 years will be a geometric series. However, we’re also told this piece
of information in the question.
We need to determine the common
ratio of this geometric series. To do this, we first need to recall
that when we say the ratio of a geometric series, we mean the ratio of the geometric
sequence which makes up this geometric series. And remember, in a geometric
sequence, to get the next term in our sequence we need to multiply it by some common
ratio we call 𝑟. We call 𝑎 the initial value of our
sequence, and we call 𝑟 the common ratio. Then when we add these together
because we’re adding terms of a geometric sequence together, we call this a
geometric series.
And remember, in this question,
we’re told the total amount of money that Chloe earns in 𝑛 years at the company is
the geometric series. So, each term in this series is
actually going to be the amount of money Chloe earns each year. There’s a few different ways of
finding this ratio. One way is to take the quotient of
two successive terms. So, let’s find two of these
successive terms. We can find the initial value by
finding the amount of money Chloe will earn in year one in her company. And we’re told this in the
question; it’s equal to 28,000 dollars.
We then want to work out how much
money she makes in the second year at the company. Remember, at this point, she will
have worked one full year in her company, so she would have had a 2.5 percent salary
increase. And there’s a few different ways of
calculating this value. For example, we could write this as
28,000 dollars plus 2.5 percent to 28,000 dollars. However, we’ll write this as 28,000
dollars multiplied by 1.025.
And this is enough to find the
common ratio. However, there is one thing worth
pointing out here. We can do exactly the same thing to
find the amount earned in year three. Once again, she’ll get a 2.5
percent salary increase for working another full year, which means we would then
need to increase the amount earned in year two by 2.5 percent. We would need to once again
multiply this by 1.025. And this is, of course, true for
any number of years. This is why this makes a geometric
series.
Now, there’s a few different ways
of finding our ratio 𝑟. For example, we could divide the
amount made in year two by the amount made in year one. However, we can also notice we’re
just multiplying by 1.025 each time. And that’s enough to answer our
question. The common ratio of this geometric
sequence 𝑟 is going to be 1.025.
The second part to this question
wants us to writes a formula for 𝑆 sub 𝑛, the total amount of dollars Chloe earns
in 𝑛 years at the company. Now, it’s worth pointing out we can
just directly answer this question by using our formula for the sum of a geometric
series. However, let’s first show why this
is true. In this case, 𝑆 sub 𝑛 is the
total amount in dollars that Chloe earns in 𝑛 years at the company. To find this value, we just need to
add together the amount she earns in year one added to the amount she earns in year
two, all the way up to the amount she would earn in year 𝑛.
In the first year, we’ve already
shown she makes 28,000 dollars. In the second year, she gets a
salary increase of 2.5 percent. So, she’ll make 28,000 multiplied
by 1.025. Adding these two together gives the
amount that she will earn in two years at the company. And, of course, we know this is
true for any number of years, so we can keep going all the way up to the amount she
will earn in year 𝑛. At this point in time, she will
have worked 𝑛 minus one full years in the company. So, she would have got 2.5 percent
salary increase 𝑛 minus one times. So, the amount she earns in year 𝑛
is 28,000 dollars multiplied by 1.025 raised to the power of 𝑛 minus one.
And just as we showed before, this
is a geometric series with initial value 𝑎, 28,000 dollars, and ratio of successive
terms 𝑟, 1.025. And we know a formula to find the
sum of the first 𝑛 terms of a geometric series. 𝑆 sub 𝑛 will be equal to 𝑎
multiplied by 𝑟 to the 𝑛th power minus one all over 𝑟 minus one. So, we substitute 𝑎 is 28,000
dollars and 𝑟 is 1.025 into this formula to get that 𝑆 sub 𝑛 is equal to 28,000
dollars multiplied by 1.025 to the 𝑛th power minus one all over 1.025 minus
one.
And all we need to do is evaluate
this expression, in our denominator, we have 1.025 minus one, which we can evaluate
is 0.025. Then all we need to do is divide
28,000 by 0.025. And if we calculate this, we get
1,120,000, which gives us our final answer. 𝑆 sub 𝑛, the total amount of
dollars Chloe will earn in 𝑛 years at the company, is equal to 1,120,000 multiplied
by 1.025 to the 𝑛th power minus one dollars.
The third part of this question
wants us to determine how much money Chloe will make if she leaves her company after
20 years. And we’re told to do this by using
our formula. This is because we could just
calculate the amount she earns in each year and then add all of these together. However, it’s far easier to use our
formula for 𝑆 sub 𝑛. Remember, since we’re finding the
amount she earns after 20 years at the company, our value of 𝑛 is going to be
20.
So, we substitute 𝑛 is equal to 20
into our formula for 𝑆 sub 𝑛 we found in the previous question. We get 𝑆 sub 20 is equal to
1,120,000 multiplied by 1.025 to the 20th power minus one dollars. And if we just calculate this
expression and give our answer to the nearest cent, we get 715,250 dollars 41
cents.
But there’s still one more part to
this question to answer, so let’s clear some space.
The last part of this question
asked us to explain why the amount she earned will be different from the amount
calculated using the formula. Option (A) she spent part of the
money in 20 years. Option (B) the value of the dollar
varies with time. Option (C) the actual amount will
have a different percentage compared to the amount calculated using the formula. Option (D) the actual amount will
have a different starting value compared to the amount calculated using the
formula. Or option (E) when necessary, the
new annual salary will be rounded.
The last part of this question
gives us an interesting problem. If Chloe were to calculate the
amount she should’ve earned by using our formula, she would find that her answer
will be different from the actual amount she earned. We’re given five possible options
as to why this will be the case. We can actually answer this
directly from our line of working. However, let’s just go through our
five options first.
Option (A) tells us that she will
have spent part of the money in 20 years. Now, while it is true, she probably
did spend part of the money in the 20 years, this will not affect the total amount
that she earned in those 20 years. All this would really affect is the
total amount of money she has left. So, option (A) is not the correct
answer.
Option (B) tells us the correct
answer should be that the total amount she earned in 20 years will be different
because the value of the dollar varies with time. And, of course, we do know it is
true that the value of the dollar will vary with time. However, for the entire 20 years
that Chloe worked for the company, she was paid in dollars. So, then no point would the value
of the dollar change the total amount of money she made because she was only ever
paid in dollars anyway. So, option (B) is not true. It won’t change the total amount of
money that she earned. However, you could make an argument
that it would change the value of the amount of money that she made, but not the
total.
Option (C) tells us that we should
have used a different percentage when we were calculating by using the formula. Once again, we know this won’t be
true because we’re told every year her salary will increase by 2.5 percent, and we
used this value throughout. So, option (C) can’t be correct
because we know her salary increases by 2.5 percent each year.
Option (D) tells us that we should
have used a different starting value for our formula. And once again, we know this is not
true because we’re told in the question that her initial starting salary is 28,000
dollars. So, after one full year in her job,
she will make 28,000 dollars. This will be the initial starting
value. So, option (D) also can’t be
correct.
This then leaves us with option
(E), which tells us, when necessary, the new annual salary will be rounded. Let’s discuss why this might change
the actual amount that she earned. And to really highlight this, let’s
calculate the amount of money she would earn in each year at her company. Let’s start with the first
year. Of course, in the first year, she
earns her starting salary 28,000 dollars. In the second year, she’ll earn the
28,000 dollars plus her salary increase of 2.5 percent. So that’s 28,000 multiplied by
1.025. And if we calculate this value,
we’ll get 28,700 dollars exactly.
And let’s do the same for year
three and year four. In year three, we’ll calculate that
she makes 28,000 multiplied by 1.025 squared and in year four 28,000 dollars
multiplied by 1.025 cubed. And if we calculate these, in year
three, her salary is 29,417 dollars 50 cents. And in year four, we get 30,152
dollars and 93 cents. But we also get an extra 0.75
cents. And this is where our problem would
start to rise because the company can’t give her 0.75 of a cent. So, most likely, the company would
round up and give her 30,152 dollars and 94 cents.
However, our formula adds together
the exact amount calculated each year, whereas the actual amount she earned would be
using the exact value she gets given. And this rounding can make our
formula incorrect in this case. And it’s worth pointing out that
this is only true when we’re working in dollars and cents because we can’t in this
case give 0.75 of a cent. But this isn’t always true. For example, if we were working in
length, then we can keep going as low as we want. This is why when working with
real-world problems, it’s very important to know all of the things you’re working
with.
We were able to show that the
amount that she earned was different to the amount we calculated using the formula
because of options (E), when necessary, the new annual salary will need to be
rounded.
Let’s now see a different example
of a real-world problem involving geometric sequences in series.
A gold mine produced 2,257
kilograms in the first year but decreased 14 percent annually. Find the total amount of gold
produced in the third year and the total across all three years. Give the answers to the nearest
kilogram.
In this question, we’re given some
information about a gold mine. We’re told that in the first year
of production, the gold mine produces 2,257 kilograms. But we’re told that year on year,
this amount is decreasing by 14 percent. The question wants us to find two
things. It wants us to find the amount of
gold which is produced in the third year of production, and it wants us to find the
total amount produced in all three of the first years. And we need to give both of our
answers to the nearest kilogram.
There’s actually two different ways
we can answer this question. The first way is to directly find
these values from the information given to us in the question. We’re told in the question, in the
first year, the gold mine produces 2,257 kilograms of gold. We can find the amount of gold
produced in the second year by remembering that this amount is going to decrease by
14 percent every year. There’s a few different ways of
evaluating a decrease of 14 percent.
One way is to multiply by one minus
0.14. And it’s worth pointing out here
we’re subtracting 0.14 because this is a decrease. So we need to subtract, and we get
0.14 because our rate, 𝑟, is 14 and we need to divide this by 100. What we’re really saying here is a
decrease in 14 percent is the same as multiplying by 0.86. Therefore, the amount of gold
produced in the mine in the second year is 2,257 multiplied by 0.86 kilograms. And we can evaluate this exactly;
we get 1941.02 kilograms. And we shouldn’t round our answer
until the very end of the question, so we’ll leave this in exact form.
We’re then going to want to do
exactly the same for the third year. Once again, from the question, we
know that the mine is going to produce 14 percent less gold in the third year than
it did in the second year. So, one thing we could do is
multiply the amount of gold we got in the second year by 0.86. However, it’s actually easy to just
multiply our expression by 0.86. Multiplying this expression by 0.86
and simplifying, we get 2,257 multiplied by 0.86 squared kilograms. Calculating this expression
exactly, we get 1669.2772 kilograms.
We can now use these three values
to answer our question. First, we can find the amount of
gold produced in the third year by rounding this number to the nearest kilogram. This would then give us 1,669
kilograms. Next, we can find the total amount
of gold produced in three years by adding these three values together. This gives us 2,257 kilograms plus
1941.02 kilograms plus 1669.2772 kilograms. And if we evaluate this expression,
we get 5867.2972 kilograms. And to the nearest kilogram, we can
see our first decimal place is two, so we need to round down, giving us 5,867
kilograms.
However, what would have happened
if we needed to find even more years of production? We can see that this method only
really worked because we only had to calculate the first three years. If we were asked to find even more
years in our example, we would need to notice something interesting. Each year we’re multiplying by a
constant ratio of 0.86. And remember, in a sequence, if
we’re multiplying by a constant ratio to get the next term in our sequence, we call
this a geometric sequence.
So, the gold produced in our mine
after 𝑛 years forms a geometric sequence with initial value 𝑎, 2,257 kilograms,
and ratio 𝑟, 0.86. We can then use what we know about
geometric sequences to find the amount of gold produced after 𝑛 years in our mine
and the total amount of gold produced after 𝑛 years. We just substitute 𝑛 is equal to
three and our values for 𝑎 and 𝑟 into the two formula to find these
expressions. And after rounding, we get the same
answers we had before. 𝑎 sub three will be 1,669
kilograms and 𝑆 sub three will be 5,867 kilograms.
Let’s now go over the key points of
this video. First, a lot of real-world problems
involve geometric sequences and series. We also know if we can turn any of
these real-world problems into a problem involving geometric sequence or series,
then we can use any of the results we know about geometric sequences or series to
help us answer these questions. Finally, we should always check
that our answers make sense in the real-world situation we’re given. For example, sometimes, our
calculations will involve noninteger values for populations. And we always need to be worried
how this might affect our final answer.
