Worksheet: Applications on Geometric Sequences

In this worksheet, we will practice solving real-world applications of geometric sequences, where we will find the common ratio, the nth term explicit formula, and the order and value of a specific sequence term.

Q1:

After falling, a rubber ball bounces back to 14 of its previous height. Given that the ball fell from a height of 653 cm above the ground, find, to the nearest integer, the height it would reach after its second bounce.

Q2:

The population of a city is now 844,501 and increases at a constant rate of 12% per year. Find the population after 8 years giving the answer to the nearest integer.

Q3:

The monthly salary of an employee this year is 2,100 LE. Find his salary after 8 years given he received a 3% raise every year, giving the answer to the nearest pound.

Q4:

The value of a brand-new car which cost 498,300 LE decreases at a rate of 15% every year. Find the price of the car after 8 years giving the answer to the nearest pound.

Q5:

On the first day, 42 liters of water are poured into a tank. Every day thereafter, three times as much water is poured into the tank as was poured on the previous day. On which day are 1,134 liters poured into the tank?

  • Aday 4
  • Bday 5
  • Cday 3
  • Dday 6
  • Eday 27

Q6:

A pendulum travels a distance of 3 feet on its first swing. On each successive swing, it travels 34 the distance of the previous swing. What is the total distance the pendulum has traveled when it stops swinging?

Q7:

A ball rebounds to π‘Ÿ times its previous height after each bounce. It is observed to rebound to a tenth of its original height on the 6th bounce. What is the value of π‘Ÿ? Round your answer to two decimal places.

Q8:

A man sent a message to two of his friends, and each of them then sent the same message to another two friends and so on. Find the number of people who got the message the sixth time it was sent given that each person received the message only once.

Q9:

The number of students in a school increases by 10% every year and there are currently 2,988 students. How many students will the school have after 6 years?

Q10:

Upon releasing a new TV show, 10 people downloaded it. If the number of people doubles every other day, how many people will be downloading the show at the end of 2 weeks ?

Q11:

William and Jennifer are investigating number sequences.

The 𝑛th term of William’s sequence can be found using the formula π‘Ž=π‘™οŠοŠ.

The 𝑛th term of Jennifer’s sequence can be found using the formula π‘Ž=π‘οŠοŠ©οŠ.

The first term of William’s sequence is 125. If Jennifer’s and William’s sequences are to be the same, what should Jennifer choose as the value of 𝑐?

Matthew wants to produce a sequence using the formula π‘Ž=25οŠο‡οŠ. If his sequence is the same as William’s and Jennifer’s, what is the value of π‘˜?

  • A32
  • B12
  • C23
  • D3
  • E13

Q12:

The count of a termite population from week to week is modeled by the recursion 𝑝=1.05π‘οŠοŠ°οŠ§οŠ. If the population after 24 weeks is 1,200, what was the initial population?

Q13:

The count of a termite population from week to week is modeled by the recursion 𝑝=1.05π‘οŠοŠ°οŠ§οŠ. What recursion would be used if the population was counted monthly? Use the average number of weeks per month over one year knowing there are 52 weeks or 12 months in a year, and approximate your answer to two decimal places.

  • A𝑝=1.24π‘οŠοŠ°οŠ§οŠ
  • B𝑝=2.54π‘οŠοŠ°οŠ§οŠ
  • C𝑝=0.26π‘οŠοŠ°οŠ§οŠ
  • D𝑝=4.20π‘οŠοŠ°οŠ§οŠ
  • E𝑝=1.26π‘οŠοŠ°οŠ§οŠ

Q14:

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.

Q15:

Victoria is training in the gym. On the treadmill, she runs 250 m in the first minute and the distance she runs decreases by 10% each subsequent minute.

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

Q16:

Olivia joined a company with a starting salary of $28,000. 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 π‘†οŠ, the total amount in dollars that Olivia earns in 𝑛 years at the company.

  • A𝑆=1,120,000(1.025βˆ’1)
  • B𝑆=58,947(1.475βˆ’1)
  • C𝑆=18,667(2.5βˆ’1)
  • D𝑆=28,000(1βˆ’1.025)
  • E𝑆=1,120,000(1βˆ’0.975)

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.

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

Q17:

When Noah first moved into his apartment his rent was $13,200 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.

Q18:

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 𝑝100).

On the day the 𝑛th deposit is made, the first deposit has been earning interest for (π‘›βˆ’1) months, so its value is 𝐷(1+𝑖).

Similarly, on the day the 𝑛th deposit is made, the second deposit has been earning interest for (π‘›βˆ’2) months, so its value is 𝐷(1+𝑖).

The pattern continues until we consider the 𝑛th 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 𝑛th deposit is made, we need to find the sum of the values of the individual deposits.

Starting with the 𝑛th deposit, we get 𝑇=𝐷+𝐷(1+𝑖)+𝐷(1+𝑖)+β‹―+𝐷(1+𝑖).

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

  • Aarithmetic
  • Bharmonic
  • Cgeometric
  • DFibonacci

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.

  • A𝑇=𝐷(1+𝑖)βˆ’1𝑖+1ο‰οŠ
  • B𝑇=𝐷(1+𝑖)βˆ’1π‘–οŠοŠ
  • C𝑇=𝐷(1+𝑖)βˆ’1π‘–οŠοŠοŠ±οŠ§
  • D𝑇=𝐷1βˆ’π‘–1βˆ’π‘–ο‰οŠ
  • E𝑇=𝐷1βˆ’π‘–1βˆ’π‘–οŠοŠοŠ±οŠ§

Q19:

Emma saves $50 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?

Q20:

Mason saves $20 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?

Q21:

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 $5,000 for the deposit on the car? Give your answer to the nearest dollar.

Q22:

A couple want to buy an apartment for $250,000. The monthly mortgage payment can be calculated using the formula 𝑃=𝐿⋅𝑖1βˆ’(1+𝑖),

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 0.275% per month, and they have a down payment of $50,000. 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 $1,000 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?

  • AYes
  • BNo

Q23:

The loan amount and the monthly payment on the loan are related by the formula 𝐿=𝑃1βˆ’(1+𝑖))𝑖, 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 $25,000, with no down payment.

Q24:

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?

Q25:

The loan amount and the monthly payment on the loan are related by the formula 𝐿=𝑃(1βˆ’(1+𝑖))𝑖, 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 $20,000, 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.

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