Lesson Explainer: Applications of Geometric Sequences and Series Mathematics

In this explainer, we will learn how to solve real-world applications of geometric sequences and series, where we will find the common ratio, the ๐‘›th term explicit formula, the order and value of a specific sequence term, and the sum of a given number of terms.

Let us consider a sequence where each term is found by multiplying the previous term by a constant. For example, the sequence 2,6,18,54,162,486,โ€ฆ.

We call this constant multiplier the common ratio. Another way to describe the sequence would be to say that each term in the sequence is equal to the previous term multiplied by the common ratio.

This is known as a geometric sequence, in this case with a first term equal to 2 and a common ratio of 3. If our sequence just consisted of, say, the six terms above (or indeed any specific number of terms), then we call it a finite geometric sequence, because it has a finite number of terms. If the sequence followed this pattern forever, as the ellipsis at the end implies, then we call it an infinite geometric sequence.

Definition: Geometric Sequence

A geometric sequence is a sequence that has a common ratio, ๐‘Ÿ, between consecutive terms. The first term is denoted by ๐‘‡ or ๐‘‡๏Šง, the second term ๐‘‡๏Šจ, the third term ๐‘‡๏Šฉ, and so on. The ๐‘›th term is denoted by ๐‘‡๏Š.

Each term is found by multiplying the previous term by the common ratio: ๐‘‡=๐‘‡,๐‘‡=๐‘‡ร—๐‘Ÿ,๐‘‡=๐‘‡ร—๐‘Ÿ,โ€ฆ,๐‘‡=๐‘‡ร—๐‘Ÿ.๏Šง๏Šจ๏Šง๏Šฉ๏Šจ๏Š๏Š๏Šฑ๏Šง

This can also be expressed as the first term multiplied by powers of the common ratio: ๐‘‡=๐‘‡,๐‘‡=๐‘‡๐‘Ÿ,๐‘‡=๐‘‡๐‘Ÿ,โ€ฆ๏Šง๏Šจ๏Šฉ๏Šจ such that the ๐‘›th term is defined by ๐‘‡=๐‘‡๐‘Ÿ๏Š๏Š๏Šฑ๏Šง.

Returning to our initial geometric sequence above, if we know the numbers in the sequence, we can calculate the common ratio by dividing the value of one term by the value of the previous term. Since the ratio is common between all successive pairs of terms, it does not matter which pair we choose for our calculation.

The ratio of the first two terms is 6รท2=3, the ratio of the second two terms is 18รท6=3, and so on.

Definition: The Common Ratio

Since we multiply one term by the common ratio to get the next term, we can express this generally as ๐‘‡=๐‘‡ร—๐‘Ÿ,๏Š๏Šฐ๏Šง๏Š and by dividing both sides of the equation by ๐‘‡๏Š, we get ๐‘Ÿ=๐‘‡๐‘‡.๏Š๏Šฐ๏Šง๏Š

Alternatively, with the definition that any term is the result of multiplying the previous term by the common ratio, we find that ๐‘Ÿ=๐‘‡๐‘‡.๏Š๏Š๏Šฑ๏Šง

The sum of the terms in a sequence is called a series. Given the geometric sequence (2,6,18,54,162,486,1458,4374,13122,39366,โ€ฆ), the corresponding geometric series could be represented as follows: 2+6+18+54+162+486+1458+4374+13122+39366+โ‹ฏ.

In this case, by adding together the first 10 terms in the series, we can see that the sum of these terms is 59โ€Žโ€‰โ€Ž048.

We will now derive a formula for the sum of the first ๐‘› terms of geometric sequence.

Consider a geometric sequence with first term ๐‘‡ and common ratio ๐‘Ÿ. The first ๐‘› terms can be written as ๐‘‡,๐‘‡๐‘Ÿ,๐‘‡๐‘Ÿ,โ€ฆ,๐‘‡๐‘Ÿ๏Šจ๏Š๏Šฑ๏Šง, so the sum of the first ๐‘› terms of a geometric sequence can be written as follows:

๐‘†=๐‘‡+๐‘‡๐‘Ÿ+๐‘‡๐‘Ÿ+โ‹ฏ+๐‘‡๐‘Ÿ+๐‘‡๐‘Ÿ.๏Š๏Šจ๏Š๏Šฑ๏Šจ๏Š๏Šฑ๏Šง(1)

If we multiply both sides of our equation by ๐‘Ÿ, we have

๐‘Ÿ๐‘†=๐‘‡๐‘Ÿ+๐‘‡๐‘Ÿ+๐‘‡๐‘Ÿ+โ‹ฏ+๐‘‡๐‘Ÿ+๐‘‡๐‘Ÿ.๏Š๏Šจ๏Šฉ๏Š๏Šฑ๏Šง๏Š(2)

When we subtract the terms in equation (2) from the terms in equation (1), all but the terms ๐‘‡ and ๐‘‡๐‘Ÿ๏Š cancel out: ๐‘†=๐‘‡+๐‘‡๐‘Ÿ+๐‘‡๐‘Ÿ+โ‹ฏ+๐‘‡๐‘Ÿ+๐‘‡๐‘Ÿ,๐‘Ÿ๐‘†=๐‘‡๐‘Ÿ+๐‘‡๐‘Ÿ+๐‘‡๐‘Ÿ+โ‹ฏ+๐‘‡๐‘Ÿ+๐‘‡๐‘Ÿ.๏Š๏Šจ๏Š๏Šฑ๏Šจ๏Š๏Šฑ๏Šง๏Š๏Šจ๏Šฉ๏Š๏Šฑ๏Šง๏Š

So, ๐‘†โˆ’๐‘Ÿ๐‘†=๐‘‡โˆ’๐‘‡๐‘Ÿ.๏Š๏Š๏Š

Factoring ๐‘†๏Š from the right-hand side and ๐‘‡ from the left-hand side will allow us to create an equation for ๐‘†๏Š: ๐‘†(1โˆ’๐‘Ÿ)=๐‘‡(1โˆ’๐‘Ÿ)๐‘†=๐‘‡(1โˆ’๐‘Ÿ)1โˆ’๐‘Ÿ.๏Š๏Š๏Š๏Š

Alternatively, we could have subtracted (1) from equation (2) to obtain the formula ๐‘†=๐‘‡(๐‘Ÿโˆ’1)๐‘Ÿโˆ’1.๏Š๏Š

Definition: The Sum of a Finite Geometric Sequence

The sum of the first ๐‘› terms of a geometric sequence, with first term ๐‘‡ and common ratio ๐‘Ÿ, is denoted by ๐‘†๏Š: ๐‘†=๐‘‡(1โˆ’๐‘Ÿ)1โˆ’๐‘Ÿ๐‘†=๐‘‡(๐‘Ÿโˆ’1)๐‘Ÿโˆ’1.๏Š๏Š๏Š๏Šor

Generally, we use the first version when ๐‘Ÿ<1 and the second one when ๐‘Ÿ>1.

If ๐‘Ÿ=1, all the terms of the geometric sequence are the same, so we would just need to multiply the first term by the number of terms: ๐‘†=๐‘‡ร—๐‘›๏Š.

We will now look at how we can apply some of the above formulae to solve real-world problems involving geometric sequences and series.

Example 1: Solving an Applied Problem Using Geometric Sequences

Amira joined a company with a starting salary of $28โ€Žโ€‰โ€Ž000. She receives a 2.5%โ€Ž salary increase after each full year in the job.

  1. The total Amira earns over ๐‘› years is a geometric series. What is the common ratio?
  2. Write a formula for ๐‘†๏Š, the total amount in dollars that Amira earns in ๐‘› years at the company.
  3. After 20 years with the company, Amira leaves. Use your formula to calculate the total amount she earned there.
  4. Explain why the actual amount she earned will be different from the amount calculated using the formula.
  1. She spent part of the money within the 20 years.
  2. The value of the dollar varies with time.
  3. When necessary, the new annual salary will be rounded.
  4. The actual amount will have a different percentage compared to the amount calculated using the formula.
  5. The actual amount will have a different starting value compared to the amount calculated using the formula.

Answer

There are four parts to this question that we will work through individually.

Part 1

We are given that Amira has a starting salary of $28โ€Žโ€‰โ€Ž000 and that she receives a salary increase of 2.5% after each full year in the job. This is enough information to determine that the amount she earns in ๐‘› years will be a geometric sequence. The first part of this question asks us to calculate the common ratio of this series.

When we talk about the ratio of a geometric series, we mean the ratio of the geometric sequence that makes up that series.

Any geometric sequence can be written in the form ๐‘‡,๐‘‡๐‘Ÿ,๐‘‡๐‘Ÿ,โ€ฆ,๐‘‡๐‘Ÿ๏Šจ๏Š๏Šฑ๏Šง, where ๐‘‡ is the first term and ๐‘Ÿ is the common ratio. We can calculate the common ratio by calculating the quotient of two successive terms: ๐‘Ÿ=๐‘‡๐‘‡.๏Š๏Šฐ๏Šง๏Š

We know that in yearโ€‰โ€‰1, Amira will earn $28โ€Žโ€‰โ€Ž000. In year 2, she will have had a 2.5% salary increase. We can therefore calculate the amount of money Amira earns in year 2 by calculating 2.5% of $28โ€Žโ€‰โ€Ž000 and then adding this value on.

An alternative method here would be to use the multiplier method. As Amiraโ€™s salary is increasing by 2.5%, we need to multiply her salary be 1.025.

In year 2, she will earn $28000ร—1.025=$28700.

This is enough to find the common ratio as we can see that the first term ๐‘‡=$28000 and the common ratio ๐‘Ÿ=1.025.

We could, however, continue this pattern to show how much money Amira earns in year 3, year 4, and so on:

  • Year1=$28000
  • Year2=$28000ร—1.025
  • Year3=$28000ร—1.025ร—1.025=$28000ร—1.025๏Šจ
  • Year4=$28000ร—1.025ร—1.025ร—1.025=$28000ร—1.025๏Šฉ
  • ...

This means that in the ๐‘›thย yearย Amira will earn $28000ร—1.025๏Š๏Šฑ๏Šง, as the exponent or power will always be 1 less than the number of years.

This ties in with our general expression for the ๐‘›th term of a geometric sequence ๐‘‡=๐‘‡๐‘Ÿ๏Š๏Š๏Šฑ๏Šง.

The common ratio of the sequence is 1.025.

Part 2

In the second part of the question, we need to write a formula for ๐‘†๏Š, the total amount of dollars that Amira earns in ๐‘› years at the company.

We know that the sum of the first ๐‘› terms of a geometric series, denoted ๐‘†๏Š, can be found using the following formula: ๐‘†=๐‘‡(๐‘Ÿโˆ’1)๐‘Ÿโˆ’1๏Š๏Š.

Substituting ๐‘‡=28000 and ๐‘Ÿ=1.025, we have ๐‘†=28000(1.025โˆ’1)1.025โˆ’1๐‘†=28000(1.025โˆ’1)0.025๐‘†=1120000(1.025โˆ’1).๏Š๏Š๏Š๏Š๏Š๏Š

The formula for ๐‘†๏Š, the total amount of dollars that Amira earns in ๐‘› years at the company, is ๐‘†=1120000(1.025โˆ’1)๏Š๏Š.

Part 3

In order to calculate the amount of money Amira will have earned after 20 years at the company, we need to substitute ๐‘›=20 into our previous answer: ๐‘†=1120000๏€น1.025โˆ’1๏…๐‘†=715250.41.๏Šจ๏Šฆ๏Šจ๏Šฆ๏Šจ๏Šฆ

When Amira leaves the company after 20 years, she will have earned $715โ€Žโ€‰โ€Ž250.41, to the nearest cent.

Part 4

The final part of the question asks us why the actual amount she earned will be different from the amount calculated using the formula. There are five possible answer options to this part of the question.

This is an interesting problem, and we will consider each option one at a time.

While it is true that Amira will probably have spent some of her money in the 20-year period, this would not affect the amount of money she earned, only the amount of money she had left. Therefore, option A is not the correct answer.

It is true that the value of the dollar would have changed over time; however, as Amira was always paid in dollars throughout this period, the value of the dollar will have no impact on the amount that Amira earns. Option B is also incorrect.

The third option presents an interesting problem that we come across regularly in mathematics: when to round our answers. When we use the formula to calculate the total amount that Amira earned, we use exact values for years 1โ€“20 and only round the answer at the end. However, in reality, the salary would be rounded in each year. For example, in year 4, Amira earned $28000ร—1.025=$30152.9375๏Šฉ. This would be rounded to the nearest cent, so Amira actually earned $30โ€Žโ€‰โ€Ž152.94. As a result, when we add these rounded values, we will obtain a slightly different value than when using the formula. This is true of any problem when dealing in currency as these values have to be rounded to two decimal places.

Options D and E suggest that there will be a different percentage and a different starting value; however, neither of these statements are true as the percentage increase in salary is always 2.5% and the starting salary is always $28โ€Žโ€‰โ€Ž000. Both of these answers are therefore incorrect.

The actual amount Amira earned will be different from the amount calculated using the formula, because when necessary, the new annual salary will be rounded. The correct answer is option C.

Example 2: Modeling a Real-World Problem Using a Geometric Sequence and Using It to Solve a Problem

A gold mine produced 2โ€Žโ€‰โ€Ž257 kg in the first year but production decreased by 14% annually. Find the amount of gold produced in the third year and the total across all 3 years. Give the answers to the nearest kilogram.

Answer

We need to calculate the amount of gold produced in the 3rd year and the total amount produced across all three years. One way of doing this would be to directly find these values from the information given in the question.

We are told that the amount of gold produced in the first year is 2โ€Žโ€‰โ€Ž257 kg.

In the second year, there is a 14% decrease. We could calculate 14% of 2โ€Žโ€‰โ€Ž257 kg and then subtract this value from 2โ€Žโ€‰โ€Ž257 kg. Alternatively, we could multiply 2โ€Žโ€‰โ€Ž257 kg by (1โˆ’0.14), as 14% written as a decimal is 0.14. This gives us a multiplier equal to 0.86.

These methods only really work when we need to calculate a small number of years.

If we needed to calculate over a longer period of time, we can use our knowledge of geometric sequences. We know that any geometric sequence has a first term ๐‘‡ and common ratio ๐‘Ÿ.

The amount of gold produced by the mine forms such a sequence, where ๐‘‡=2257 and ๐‘Ÿ=0.86. We know that the common ratio, ๐‘Ÿ, is 0.86 as this is the constant that we multiply each term by to get the next term.

The general term of a geometric sequence, ๐‘‡๏Š, can be calculated using the formula ๐‘‡=๐‘‡๐‘Ÿ๏Š๏Š๏Šฑ๏Šง. Substituting in our values, we have ๐‘‡=2257(0.86)=2257(0.86)=1669.2772.๏Šฉ๏Šฉ๏Šฑ๏Šง๏Šจ

Once again, we see that the amount of gold produced in the third year is equal to 1โ€Žโ€‰โ€Ž669 kg, rounded to the nearest kilogram.

The sum of the first ๐‘› terms of a geometric sequence ๐‘†๏Š can be calculated using the formula ๐‘†=๐‘‡(1โˆ’๐‘Ÿ)1โˆ’๐‘Ÿ๏Š๏Š. Substituting in our values, we have ๐‘†=2257๏€น1โˆ’0.86๏…1โˆ’0.86๐‘†=2257๏€น1โˆ’0.86๏…0.14๐‘†=5867.2972.๏Šฉ๏Šฉ๏Šฉ๏Šฉ๏Šฉ

The total amount of gold produced across all three years is equal to 5โ€Žโ€‰โ€Ž867 kg, rounded to the nearest kilogram.

In the next example, we will consider the situation when an amount of money is invested into a savings account where an annual interest rate is compounded monthly.

Example 3: Solving a Finance Problem Where Interest Is Compounded Monthly Using Geometric Sequences

Sameh saves $20 every month in an account that pays an annual interest rate of 4% compounded monthly.

  1. How much will be in Samehโ€™s account after 4 years of regular saving? Give your answer to the nearest cent.
  2. If the interest was compounded quarterly, how much would be in the account after 4 years?

Answer

There are two parts to this question both of which can be modeled using geometric sequences.

Part 1

Firstly, we have an account that pays an annual interest rate of 4% compounded monthly, so the monthly rate can be calculated by dividing 4% by 12: monthlyinterestrate=0.0412=1300.

The multiplier will therefore be equal to 1+1300=301300, so the common ratio ๐‘Ÿ=301300.

Sameh saves $20 every month, so the first term of the geometric sequence ๐‘‡=$20.

Over the four-year period, there will be 4ร—12=48 monthly payments, which means there are 48 terms in our geometric sequence, so ๐‘›=48.

The sum of the first ๐‘› terms of a geometric sequence, ๐‘†๏Š, can be calculated using the formula ๐‘†=๐‘‡(๐‘Ÿโˆ’1)๐‘Ÿโˆ’1๏Š๏Š. Substituting in our values, we have ๐‘†=20๏€ผโˆ’1๏ˆโˆ’1๐‘†=1039.1920โ€ฆ.๏Šช๏Šฎ๏Šฉ๏Šฆ๏Šง๏Šฉ๏Šฆ๏Šฆ๏Šช๏Šฎ๏Šฉ๏Šฆ๏Šง๏Šฉ๏Šฆ๏Šฆ๏Šช๏Šฎ

Rounding this to two decimal places, we can conclude that there is $1โ€Žโ€‰โ€Ž039.19 in Samehโ€™s account after 4 years.

Part 2

Secondly, we have an account that pays an annual interest rate of 4% compounded quarterly, so the quarterly rate can be calculated by dividing 4% by 4: quarterlyinterestrate=0.044=1100.

The multiplier will therefore be equal to 1+1100=101100, so the common ratio ๐‘Ÿ=101100.

Sameh saves $20 every month, so each quarter he will have saved 3ร—$20=$60; therefore, the first term of the geometric sequence ๐‘‡=$60.

Over the four-year period there will be 16 quarterly payments, which means there are 16 terms in our geometric sequence, so ๐‘›=16.

The sum of the first ๐‘› terms of a geometric sequence, ๐‘†๏Š, can be calculated using the formula ๐‘†=๐‘‡(๐‘Ÿโˆ’1)๐‘Ÿโˆ’1๏Š๏Š. Substituting in our values, we have ๐‘†=60๏€ผโˆ’1๏ˆโˆ’1๐‘†=1035.4718โ€ฆ.๏Šง๏Šฌ๏Šง๏Šฆ๏Šง๏Šง๏Šฆ๏Šฆ๏Šง๏Šฌ๏Šง๏Šฆ๏Šง๏Šง๏Šฆ๏Šฆ๏Šง๏Šฌ

Rounding this to two decimal places, we can conclude that there is $1โ€Žโ€‰โ€Ž035.47 in Samehโ€™s account after 4 years.

We can therefore conclude that if the interest is compounded monthly rather than quarterly, then Sameh will earn more interest across the four-year term.

In our final example we will solve another real-world problem involving geometric sequences.

Example 4: Solving a Physical Problem Involving Volume Using Geometric Sequences

A water tank had 1โ€Žโ€‰โ€Ž778 litres of water. The volume of the water decreased by 14, 28, and 56 litres over the next three days respectively. How long will it take the tank to be empty given that the water volume decreases following the same pattern?

Answer

We notice that the values 14,28,56,โ€ฆ form a geometric sequence, with first term ๐‘‡=14 and common ratio ๐‘Ÿ=2. To check this, we divide each term by the term before it: 56รท28=28รท14=2.

The sum of the first ๐‘› terms of a geometric sequence, ๐‘†๏Š, can be calculated using the formula ๐‘†=๐‘‡(๐‘Ÿโˆ’1)๐‘Ÿโˆ’1๏Š๏Š.

Since the total amount of water in the tank is 1โ€Žโ€‰โ€Ž778 litres, then ๐‘†=1778๏Š and we want to calculate the time period, ๐‘›, in days.

Substituting in our values we have 1778=14(2โˆ’1)2โˆ’11778=14(2โˆ’1)(14)127=2โˆ’1(1)128=2.๏Š๏Š๏Š๏Šdividingbothsidesbyaddingtobothsides

We know that 128 is a power of 2, so ๐‘› is an integer value.

In fact, 2๏Šญ is 128, so ๐‘›=7.

Note that this can also be solved using logarithms, although this is outside the scope of this explainer.

Therefore, the water tank will be empty after 7 days.

We can verify this answer by calculating the amount of water in the tank at the end of each day by subtracting 14,28,56,โ€ฆ individually.

End of day 1: 1778โˆ’14=1764

End of day 2: 1764โˆ’28=1736

End of day 3: 1736โˆ’56=1680

End of day 4: 1680โˆ’112=1568

End of day 5: 1568โˆ’224=1344

End of day 6: 1344โˆ’448=896

End of day 7: 896โˆ’896=0

This confirms that the water tank will be empty after 7 days.

We will finish this explainer by recapping some of the key points.

Key Points

  • Many real-world problems involve geometric sequences and series. The following definitions can help us solve these problems.
  • A finite geometric sequence has the form ๐‘‡,๐‘‡๐‘Ÿ,๐‘‡๐‘Ÿ,โ€ฆ,๐‘‡๐‘Ÿ๏Šจ๏Š๏Šฑ๏Šง, where ๐‘‡ is the first term, ๐‘Ÿ is the common ratio, and ๐‘› is the number of terms in the sequence.
  • The ๐‘›th term of a geometric sequence is ๐‘‡=๐‘‡๐‘Ÿ๏Š๏Š๏Šฑ๏Šง.
  • The common ratio ๐‘Ÿ of a geometric sequence whose ๐‘›th term is ๐‘‡๏Š is given by ๐‘Ÿ=๐‘‡๐‘‡๏Š๏Šฐ๏Šง๏Š or ๐‘Ÿ=๐‘‡๐‘‡๏Š๏Š๏Šฑ๏Šง.
  • The sum of the terms in a sequence is called a series.
  • The sum of the first ๐‘› terms of a geometric sequence, with first term ๐‘‡ and common ratio ๐‘Ÿ, is denoted by ๐‘†๏Š, where ๐‘†=๐‘‡(1โˆ’๐‘Ÿ)1โˆ’๐‘Ÿ๐‘†=๐‘‡(๐‘Ÿโˆ’1)๐‘Ÿโˆ’1.๏Š๏Š๏Š๏Šor

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