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Worksheet: Geometric Series

In this worksheet, we will practice deriving the formula to calculate any geometric series and using it to solve real-world problems.

Q1:

The sum of the terms of a sequence is called a series.

A geometric series is the sum of a geometric sequence; a geometric series with 𝑛 terms can be written as where π‘Ž is the first term and π‘Ÿ is the common ratio (the number you multiply one term by to get the next term in the sequence, π‘Ÿ β‰  1 ).

Find the sum of the first 6 terms of a geometric series with π‘Ž = 2 4 and π‘Ÿ = 1 2 .

  • A204
  • B 2 3 1 4
  • C 8 1 2
  • D 4 7 1 4
  • E 1 1 5 8

Q2:

On the first day, there were 810 bacteria on a petri dish. Find, giving the answer to the nearest integer, the bacterial count after six days given the bacteria replicate daily.

  • A 102 870 bacteria
  • B 25 110 bacteria
  • C 52 650 bacteria
  • D 51 030 bacteria
  • E 1 595 bacteria

Q3:

A water tank had 1 778 liters of water. The volume of the water decreased by 14, 28, and 56 over the next three days, respectively. How long will it take the tank to be empty given the water volume decreases at the same rate?

Q4:

We can find a formula for the sum of a geometric series. Consider the series

Multiply the expression for 𝑆 𝑛 by π‘Ÿ , the common ratio.

  • A π‘Ÿ 𝑆 = π‘Ž + π‘Ž π‘Ÿ + π‘Ž π‘Ÿ + π‘Ž π‘Ÿ + β‹― + π‘Ž π‘Ÿ 𝑛 2 3 𝑛
  • B π‘Ÿ 𝑆 = π‘Ž + π‘Ž π‘Ÿ + π‘Ž π‘Ÿ + π‘Ž π‘Ÿ + β‹― + π‘Ž π‘Ÿ 𝑛 2 3 𝑛 βˆ’ 1
  • C π‘Ÿ 𝑆 = π‘Ž π‘Ÿ + π‘Ž π‘Ÿ + π‘Ž π‘Ÿ + π‘Ž π‘Ÿ + β‹― + π‘Ž π‘Ÿ 𝑛 2 3 4 𝑛 βˆ’ 1
  • D π‘Ÿ 𝑆 = π‘Ž π‘Ÿ + π‘Ž π‘Ÿ + π‘Ž π‘Ÿ + π‘Ž π‘Ÿ + β‹― + π‘Ž π‘Ÿ 𝑛 2 3 4 𝑛
  • E π‘Ÿ 𝑆 = π‘Ž + π‘Ž π‘Ÿ + π‘Ž π‘Ÿ + π‘Ž π‘Ÿ + β‹― + π‘Ž π‘Ÿ 𝑛 2 3 4 𝑛 βˆ’ 1

So, we have and

The right-hand sides of the equations are very similar. Identify the terms that do \textbf{not} appear on the right-hand side of both equations.

  • A π‘Ž , π‘Ž π‘Ÿ 𝑛
  • B π‘Ž , π‘Ž π‘Ÿ 4
  • C π‘Ž π‘Ÿ , π‘Ž π‘Ÿ 𝑛 βˆ’ 1 𝑛
  • D π‘Ž , π‘Ž π‘Ÿ 𝑛 βˆ’ 1
  • E π‘Ž π‘Ÿ , π‘Ž π‘Ÿ 4 𝑛 βˆ’ 1

Now, consider the subtraction

Use the answer to the previous part to simplify the subtraction 𝑆 βˆ’ π‘Ÿ 𝑆 𝑛 𝑛 .

  • A 𝑆 βˆ’ π‘Ÿ 𝑆 = π‘Ž π‘Ÿ βˆ’ π‘Ž π‘Ÿ 𝑛 𝑛 𝑛 βˆ’ 1 4
  • B 𝑆 βˆ’ π‘Ÿ 𝑆 = π‘Ž βˆ’ π‘Ž π‘Ÿ 𝑛 𝑛 𝑛 βˆ’ 1
  • C 𝑆 βˆ’ π‘Ÿ 𝑆 = π‘Ž βˆ’ π‘Ž π‘Ÿ 𝑛 𝑛 𝑛
  • D 𝑆 βˆ’ π‘Ÿ 𝑆 = π‘Ž π‘Ÿ βˆ’ π‘Ž π‘Ÿ 𝑛 𝑛 𝑛 βˆ’ 1 𝑛
  • E 𝑆 βˆ’ π‘Ÿ 𝑆 = π‘Ž βˆ’ π‘Ž π‘Ÿ 𝑛 𝑛 4

Factor both sides of the equation.

  • A 𝑆 ( 1 βˆ’ π‘Ÿ ) = π‘Ž ( 1 βˆ’ π‘Ÿ ) 𝑛 𝑛
  • B 𝑆 ( 1 βˆ’ π‘Ÿ ) = π‘Ž ο€Ή π‘Ÿ βˆ’ π‘Ÿ  𝑛 𝑛 βˆ’ 1 4
  • C 𝑆 ( 1 βˆ’ π‘Ÿ ) = π‘Ž ο€Ή 1 βˆ’ π‘Ÿ  𝑛 4
  • D 𝑆 ( 1 βˆ’ π‘Ÿ ) = π‘Ž ο€Ή 1 βˆ’ π‘Ÿ  𝑛 𝑛 βˆ’ 1
  • E 𝑆 ( 1 βˆ’ π‘Ÿ ) = π‘Ž π‘Ÿ ( 1 βˆ’ π‘Ÿ ) 𝑛 𝑛 βˆ’ 1

Rearrange the equation to make 𝑆 𝑛 the subject of the formula.

  • A 𝑆 = π‘Ž ο€Ή 1 βˆ’ π‘Ÿ  1 βˆ’ π‘Ÿ 𝑛 4
  • B 𝑆 = π‘Ž π‘Ÿ ( 1 βˆ’ π‘Ÿ ) 1 βˆ’ π‘Ÿ 𝑛 𝑛 βˆ’ 1 𝑛
  • C 𝑆 = π‘Ž ο€Ή 1 βˆ’ π‘Ÿ  1 βˆ’ π‘Ÿ 𝑛 𝑛 βˆ’ 1
  • D 𝑆 = π‘Ž ( 1 βˆ’ π‘Ÿ ) 1 βˆ’ π‘Ÿ 𝑛 𝑛
  • E 𝑆 = π‘Ž ο€Ή π‘Ÿ βˆ’ π‘Ÿ  1 βˆ’ π‘Ÿ 𝑛 𝑛 βˆ’ 1 4

Q5:

The loan amount and the monthly payment on the loan are related by the formula where 𝐿 is the loan amount, 𝑃 is the monthly payment, 𝑖 is the monthly interest rate, and 𝑛 is the number of months over which the loan will be repaid.

A kitchen dealer is offering 6-year loans with a monthly interest rate of 0 . 4 % .

Use the formula to calculate, to the nearest cent, the monthly payment on a kitchen costing $ 2 0 0 0 0 , with no down payment.

A customer who wishes to buy the kitchen can afford payments of $250 per month. Calculate the down payment they must make for the monthly payments to be affordable. Give your answer to a suitable degree of accuracy.

Q6:

An equilateral triangle has a side length of 14 cm, where another triangle is drawn inside of it by connecting the midpoints of its sides. More interior triangles are to be repeatedly drawn the same way as shown in the figure. Find the sum of the perimeters of the first 6 triangles drawn giving the answer to the nearest integer.

Q7:

A rectangle whose length is 64 cm and width is 48 cm has its sides bisected. These points are then connected creating a rhombus. The sides of the rhombus are bisected and so on forming the figure below. Find the sum to infinity of the perimeters of the figure.

Q8:

A geometric sequence is a list of terms which can be written in the form where π‘Ž is the first term and π‘Ÿ is the common ratio (the number you multiply one term by to get the next term in the sequence, π‘Ÿ β‰  1 ).

Identify π‘Ž and π‘Ÿ in the following sequence: 4 , 1 2 , 3 6 , 1 0 8 , … .

  • A π‘Ž = 3 , π‘Ÿ = 4
  • B π‘Ž = 4 , π‘Ÿ = 8
  • C π‘Ž = 8 , π‘Ÿ = 4
  • D π‘Ž = 4 , π‘Ÿ = 3
  • E π‘Ž = 2 , π‘Ÿ = 3

Q9:

A geometric series has a first term of 3 and a common ratio of 5. Find the sum of the first 6 terms.

Q10:

To calculate the amount of money in a structured savings account, where a saver deposits a regular amount at regular time intervals, we consider each month’s deposit separately.

Consider a saver who makes a regular deposit on the last day of every month in an account where the interest is calculated on the last day of every month.

Let the regular deposit be 𝐷 , and let the monthly interest rate be 𝑖 (an interest rate of 𝑝 % would give an 𝑖 value of 𝑝 1 0 0 ).

On the day the 𝑛 th deposit is made, the first deposit has been earning interest for ( 𝑛 βˆ’ 1 ) months, so its value is 𝐷 ( 1 + 𝑖 ) 𝑛 βˆ’ 1 .

Similarly, on the day the 𝑛 th deposit is made, the second deposit has been earning interest for ( 𝑛 βˆ’ 2 ) months, so its value is 𝐷 ( 1 + 𝑖 ) 𝑛 βˆ’ 2 .

The pattern continues until we consider the 𝑛 th deposit which has earned no interest on the day it is deposited so its value is 𝐷 .

To calculate the total amount in the fund, 𝑇 , on the day the 𝑛 th deposit is made, we need to find the sum of the values of the individual deposits.

Starting with the 𝑛 th deposit, we get

What kind of series do you see on the right-hand side of the equation?

  • AFibonacci
  • Barithmetic
  • Charmonic
  • Dgeometric

Using the formula for the sum of the first 𝑛 terms of a geometric series, write a formula for 𝑇 , the total amount in the fund.

  • A 𝑇 = 𝐷 ο€Ύ ( 1 + 𝑖 ) βˆ’ 1 𝑖  𝑛
  • B 𝑇 = 𝐷 ο€Ύ ( 1 + 𝑖 ) βˆ’ 1 𝑖  𝑛 βˆ’ 1
  • C 𝑇 = 𝐷 ο€½ ( 1 + 𝑖 ) βˆ’ 1 𝑖 + 1  𝑛
  • D 𝑇 = 𝐷 ο€½ 1 βˆ’ 𝑖 1 βˆ’ 𝑖  𝑛
  • E 𝑇 = 𝐷 ο€Ύ 1 βˆ’ 𝑖 1 βˆ’ 𝑖  𝑛 βˆ’ 1

Q11:

Find, to two decimal places, the sum of the following geometric series:

Q12:

A couple want to buy an apartment for $ 2 5 0 0 0 0 . The monthly mortgage payment can be calculated using the formula

where 𝑃 is the monthly payment, 𝐿 is the loan amount, 𝑖 is the monthly interest rate, and 𝑛 is the number of months over which the mortgage will be repaid.

The bank offers a 20-year mortgage with an interest rate of 0 . 2 7 5 % per month, and they have a down payment of $ 5 0 0 0 0 . Calculate the monthly payment to the nearest cent.

What should the down payment be, to the nearest 100 dollars, if the couple can afford to pay only $ 1 0 0 0 per month?

Instead of increasing the down payment, they decide to increase the mortgage term. Assuming the same rate of interest, can they afford to make the monthly payments on a 25-year mortgage?

  • ANo
  • BYes

Q13:

Mason saves $ 2 0 every month in an account that pays an annual interest rate of 4 % compounded monthly.

How much will be in Mason’s account after 4 years of regular saving? Give your answer to the nearest cent.

If the interest was compounded quarterly, how much would be in the account after 4 years?

Q14:

A ball was rolled on a horizontal plane. It travelled 145 cm in the first minute and the distance decreased by 2 5 % every minute thereafter. Find the total distance the ball covered until it completely stopped.

Q15:

Emma saves $ 5 0 every month in a high-performing investment fund. The fund is guaranteed to pay 6 % annual interest, compounded monthly.

How much is Emma guaranteed to have in her fund at the end of 2 years?

Q16:

In a geometric sequence, the first term is π‘Ž and the common ratio is π‘Ÿ .

Find the sum of the first 3 terms of a geometric sequence with π‘Ž = 3 2 8 and π‘Ÿ = 1 4 .

  • A 3 6 9 2
  • B 5 3 3 2
  • C410
  • D 8 6 1 2

Q17:

A geometric sequence is a list of terms which can be written in the form where π‘Ž is the first term and π‘Ÿ is the common ratio (the number you multiply one term by to get the next term in the sequence, π‘Ÿ β‰  1 ).

Identify π‘Ž and π‘Ÿ in the following sequence: 2 5 0 , 5 0 , 1 0 , 2 , … .

  • A π‘Ž = 2 5 0 , π‘Ÿ = 5
  • B π‘Ž = 5 0 , π‘Ÿ = 5
  • C π‘Ž = 5 0 , π‘Ÿ = 1 0
  • D π‘Ž = 2 5 0 , π‘Ÿ = 1 5
  • E π‘Ž = 2 0 0 , π‘Ÿ = 4 5

Q18:

The loan amount and the monthly payment on the loan are related by the formula where 𝐿 is the loan amount, 𝑃 is the monthly payment, 𝑖 is the monthly interest rate, and 𝑛 is the number of months over which the loan will be repaid.

A car dealer is offering 4-year loans with a monthly interest rate of 0 . 5 % .

Use the formula to calculate the monthly payment on a car costing $ 2 5 0 0 0 , with no down payment.

Q19:

When Noah first moved into his apartment his rent was $ 1 3 2 0 0 a year. Every year, the landlord has increased the rent by 3 % . Noah has been living in the apartment for 17 years. By considering the total rent paid as a geometric series, calculate the total amount of rent Noah has paid over the 17 years he has lived in the apartment. Give your answer to the nearest dollar.

Q20:

Victoria is training in the gym. On the treadmill, she runs 250 m in the first minute and the distance she runs decreases by 1 0 % each subsequent minute.

How far does she run in 10 minutes? Give your answer to the nearest meter.

Q21:

Find the fourth term of the geometric sequence given by where is the sum of the first terms.

Q22:

Hannah wants to replace her car in 2 years’ time. She decides to save some money every month, and the best savings account has an annual interest rate of 9 % compounded monthly.

How much should the regular monthly payment be if Hannah intends to save $ 5 0 0 0 for the deposit on the car? Give your answer to the nearest dollar.

Q23:

Find the geometric sequence given the first term is 324, the last term is 4, and the sum of all the terms is 484.

  • A 3 2 4 , βˆ’ 1 0 8 , 3 6 , … , 4
  • B 3 2 4 , 9 7 2 , 2 9 1 6 , … , 4
  • C 3 2 4 , βˆ’ 9 7 2 , 2 9 1 6 , … , 4
  • D 3 2 4 , 1 0 8 , 3 6 , … , 4

Q24:

There are two geometric series with a first term of 3 and whose first 3 terms have a sum of 21.

What are their common ratios?

  • A3, 7
  • B βˆ’ 2 , 3
  • C βˆ’ 2 , βˆ’ 3
  • D2, βˆ’ 3
  • E βˆ’ 3 , βˆ’ 7

Write an expression for the sum of the first 𝑛 terms of the sequence with first term 3 and common ratio 2.

  • A 3 ο€Ή 2  𝑛 βˆ’ 1
  • B 3 ο€Ή 2 βˆ’ 1  𝑛 βˆ’ 1
  • C 2 ( 1 βˆ’ 3 ) 𝑛
  • D 2 ο€Ή 3  𝑛 βˆ’ 1
  • E 2 ο€Ή 3 βˆ’ 1  𝑛 βˆ’ 1

Q25:

Olivia joined a company with a starting salary of $ 2 8 0 0 0 . She receives a 2 . 5 % salary increase after each full year in the job.

The total Olivia earns over 𝑛 years is a geometric series. What is the common ratio?

Write a formula for 𝑆  , the total amount in dollars that Olivia earns in 𝑛 years at the company.

  • A 𝑆 = 1 , 1 2 0 , 0 0 0 ( 1 . 0 2 5 βˆ’ 1 )  
  • B 𝑆 = 2 8 0 0 0 ( 1 βˆ’ 1 . 0 2 5 )  
  • C 𝑆 = 1 1 2 0 0 0 0 ( 1 βˆ’ 0 . 9 7 5 )  
  • D 𝑆 = 1 8 6 6 7 ( 2 . 5 βˆ’ 1 )  
  • E 𝑆 = 5 8 9 4 7 ( 1 . 4 7 5 βˆ’ 1 )  

After 20 years with the company, Olivia leaves. Use your formula to calculate the total amount she earned there.

Explain why the actual amount she earned will be different from the amount calculated using the formula.

  • AWhen necessary, the new annual salary will be rounded.
  • BThe actual amount will have a different starting value compared to the amount calculated using the formula.
  • CThe actual amount will have a different percentage compared to the amount calculated using the formula.
  • DShe spent part of the money within the 20 years.
  • EThe value of the dollar varies with time.

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