1.04k Modelling with sequences: compound interest, growth/decay

8 A woman's basic salary for her first year with a particular company is \(\\) 30000\( and at the end of the year she also gets a bonus of \)\\( 600\).

1. For her first year, express her bonus as a percentage of her basic salary.
   At the end of each complete year, the woman's basic salary will increase by \(3 \%\) and her bonus will increase by \(\\) 100$.
2. Express the bonus she will be paid at the end of her 24th year as a percentage of the basic salary paid during that year.

3 Each year a company gives a grant to a charity. The amount given each year increases by \(5 \%\) of its value in the preceding year. The grant in 2001 was \(\\) 5000$. Find

1. the grant given in 2011,
2. the total amount of money given to the charity during the years 2001 to 2011 inclusive.

8 A television quiz show takes place every day. On day 1 the prize money is \(\\) 1000$. If this is not won the prize money is increased for day 2 . The prize money is increased in a similar way every day until it is won. The television company considered the following two different models for increasing the prize money. Model 1: Increase the prize money by \(\\) 1000$ each day.
Model 2: Increase the prize money by \(10 \%\) each day.
On each day that the prize money is not won the television company makes a donation to charity. The amount donated is \(5 \%\) of the value of the prize on that day. After 40 days the prize money has still not been won. Calculate the total amount donated to charity

1. if Model 1 is used,
2. if Model 2 is used.

3 A company producing salt from sea water changed to a new process. The amount of salt obtained each week increased by \(2 \%\) of the amount obtained in the preceding week. It is given that in the first week after the change the company obtained 8000 kg of salt.

1. Find the amount of salt obtained in the 12th week after the change.
2. Find the total amount of salt obtained in the first 12 weeks after the change.

8

1. The third and fourth terms of a geometric progression are 48 and 32 respectively. Find the sum to infinity of the progression.
2. Two schemes are proposed for increasing the amount of household waste that is recycled each week. Scheme \(A\) is to increase the amount of waste recycled each month by 0.16 tonnes.
   Scheme \(B\) is to increase the amount of waste recycled each month by \(6 \%\) of the amount recycled in the previous month.
   The proposal is to operate the scheme for a period of 24 months. The amount recycled in the first month is 2.5 tonnes. For each scheme, find the total amount of waste that would be recycled over the 24 -month period. Scheme \(A\) Scheme \(B\) \(\_\_\_\_\)

3

1. Each year, the value of a certain rare stamp increases by \(5 \%\) of its value at the beginning of the year. A collector bought the stamp for \(\\) 10000\( at the beginning of 2005. Find its value at the beginning of 2015 correct to the nearest \)\\( 100\).
2. The sum of the first \(n\) terms of an arithmetic progression is \(\frac { 1 } { 2 } n ( 3 n + 7 )\). Find the 1 st term and the common difference of the progression.

4 A runner who is training for a long-distance race plans to run increasing distances each day for 21 days. She will run \(x \mathrm {~km}\) on day 1 , and on each subsequent day she will increase the distance by \(10 \%\) of the previous day's distance. On day 21 she will run 20 km .

1. Find the distance she must run on day 1 in order to achieve this. Give your answer in km correct to 1 decimal place.
2. Find the total distance she runs over the 21 days.

12. A business is expected to have a yearly profit of \(\pounds 275000\) for the year 2016. The profit is expected to increase by \(10 \%\) per year, so that the expected yearly profits form a geometric sequence with common ratio 1.1

1. Show that the difference between the expected profit for the year 2020 and the expected profit for the year 2021 is \(\pounds 40300\) to the nearest hundred pounds.
2. Find the first year for which the expected yearly profit is more than one million pounds.
3. Find the total expected profits for the years 2016 to 2026 inclusive, giving your answer to the nearest hundred pounds.

11. Wheat is to be grown on a farm. A model predicts that the mass of wheat harvested on the farm will increase by \(1.5 \%\) per year, so that the mass of wheat harvested each year forms a geometric sequence. Given that the mass of wheat harvested during year one is 6000 tonnes,

1. show that, according to the model, the mass of wheat harvested on the farm during year 4 will be approximately 6274 tonnes. During year \(N\), according to the model, there is predicted to be more than 8000 tonnes of wheat harvested on the farm.
2. Find the smallest possible value of \(N\). It costs \(\pounds 5\) per tonne to harvest the wheat.
3. Assuming the model, find the total amount that it would cost to harvest the wheat from year one to year 10 inclusive. Give your answer to the nearest \(\pounds 1000\).

9. A cyclist aims to travel a total of 1200 km over a number of days. He cycles 12 km on day 1
He increases the distance that he cycles each day by \(6 \%\) of the distance cycled on the previous day, until he reaches the total of 1200 km .

1. Show that on day 8 he cycles approximately 18 km . He reaches his total of 1200 km on day \(N\), where \(N\) is a positive integer.
2. Find the value of \(N\). The cyclist stops when he reaches 1200 km .
3. Find the distance that he cycles on day \(N\). Give your answer to the nearest km .

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

In an arithmetic series,

• the sixth term is 2
• the sum of the first ten terms is - 80

For this series,

1. find the value of the first term and the value of the common difference.
2. Hence find the smallest value of \(n\) for which $$S _ { n } > 8000$$

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

The number of dormice and the number of voles on an island are being monitored.
Initially there are 2000 dormice on the island.
A model predicts that the number of dormice will increase by \(3 \%\) each year, so that the numbers of dormice on the island at the end of each year form a geometric sequence.

1. Find, according to the model, the number of dormice on the island 6 years after monitoring began. Give your answer to 3 significant figures. The number of voles on the island is being monitored over the same period of time.
   Given that
   • 4 years after monitoring began there were 3690 voles on the island
   • 7 years after monitoring began there were 3470 voles on the island
   • the number of voles on the island at the end of each year is modelled as a geometric sequence
   • find the equation of this model in the form
   $$N = a b ^ { t }$$ where \(N\) is the number of voles, \(t\) years after monitoring began and \(a\) and \(b\) are constants. Give the value of \(a\) and the value of \(b\) to 2 significant figures. When \(t = T\), the number of dormice on the island is equal to the number of voles on the island.
2. Find, according to the models, the value of \(T\), giving your answer to one decimal place.

2. The adult population of a town at the start of 2019 is 25000 A model predicts that the adult population will increase by \(2 \%\) each year, so that the number of adults in the population at the start of each year following 2019 will form a geometric sequence.

1. Find, according to the model, the adult population of the town at the start of 2032 It is also modelled that every member of the adult population gives \(\pounds 5\) to local charity at the start of each year.
2. Find, according to these models, the total amount of money that would be given to local charity by the adult population of the town from 2019 to 2032 inclusive. Give your answer to the nearest \(\pounds 1000\)

6. A car was purchased for \(\pounds 18000\) on 1 st January. On 1st January each following year, the value of the car is \(80 \%\) of its value on 1st January in the previous year.

1. Show that the value of the car exactly 3 years after it was purchased is \(\pounds 9216\). The value of the car falls below \(\pounds 1000\) for the first time \(n\) years after it was purchased.
2. Find the value of \(n\). An insurance company has a scheme to cover the maintenance of the car. The cost is \(\pounds 200\) for the first year, and for every following year the cost increases by \(12 \%\) so that for the 3rd year the cost of the scheme is \(\pounds 250.88\)
3. Find the cost of the scheme for the 5th year, giving your answer to the nearest penny.
4. Find the total cost of the insurance scheme for the first 15 years.
   \section*{LU}

9. The adult population of a town is 25000 at the end of Year 1. A model predicts that the adult population of the town will increase by \(3 \%\) each year, forming a geometric sequence.

1. Show that the predicted adult population at the end of Year 2 is 25750.
2. Write down the common ratio of the geometric sequence. The model predicts that Year \(N\) will be the first year in which the adult population of the town exceeds 40000.
3. Show that $$( N - 1 ) \log 1.03 > \log 1.6$$
4. Find the value of \(N\). At the end of each year, each member of the adult population of the town will give \(\pounds 1\) to a charity fund. Assuming the population model,
5. find the total amount that will be given to the charity fund for the 10 years from the end of Year 1 to the end of Year 10, giving your answer to the nearest \(\pounds 1000\).

6. At the beginning of the year 2000 a company bought a new machine for \(\pounds 15000\). Each year the value of the machine decreases by \(20 \%\) of its value at the start of the year.

1. Show that at the start of the year 2002, the value of the machine was \(\pounds 9600\). When the value of the machine falls below \(\pounds 500\), the company will replace it.
2. Find the year in which the machine will be replaced. To plan for a replacement machine, the company pays \(\pounds 1000\) at the start of each year into a savings account. The account pays interest at a fixed rate of \(5 \%\) per annum. The first payment was made when the machine was first bought and the last payment will be made at the start of the year in which the machine is replaced.
3. Using your answer to part (b), find how much the savings account will be worth immediately after the payment at the start of the year in which the machine is replaced.

4 There is a flowerhead at the end of each stem of an oleander plant. The next year, each flowerhead is replaced by three stems and flowerheads, as shown in Fig. 11. \begin{figure}[h] \end{figure}

1. How many flowerheads are there in year 5 ?
2. How many flowerheads are there in year \(n\) ?
3. As shown in Fig. 11, the total number of stems in year 2 is 4, (that is, 1 old one and 3 new ones). Similarly, the total number of stems in year 3 is 13 , (that is, \(1 + 3 + 9\) ). Show that the total number of stems in year \(n\) is given by \(\frac { 3 ^ { n } - 1 } { 2 }\).
4. Kitty's oleander has a total of 364 stems. Find
   (A) its age,
   (B) how many flowerheads it has.
5. Abdul's oleander has over 900 flowerheads. Show that its age, \(y\) years, satisfies the inequality \(y > \frac { \log _ { 10 } 900 } { \log _ { 10 } 3 } + 1\).
   Find the smallest integer value of \(y\) for which this is true.

5 Jim and Mary are each planning monthly repayments for money they want to borrow.

1. Jim's first payment is \(\pounds 500\), and he plans to pay \(\pounds 10\) less each month, so that his second payment is \(\pounds 490\), his third is \(\pounds 480\), and so on.
   (A) Calculate his 12th payment.
   (B) He plans to make 24 payments altogether. Show that he pays \(\pounds 9240\) in total.
2. Mary's first payment is \(\pounds 460\) and she plans to pay \(2 \%\) less each month than the previous month, so that her second payment is \(\pounds 450.80\), her third is \(\pounds 441.784\), and so on.
   (A) Calculate her 12th payment.
   (B) Show that Jim's 20th payment is less than Mary's 20th payment but that his 19th is not less than her 19th.
   (C) Mary plans to make 24 payments altogether. Calculate how much she pays in total.
   (D) How much would Mary's first payment need to be if she wishes to pay \(2 \%\) less each month as before, but to pay the same in total as Jim, \(\pounds 9240\), over the 24 months?

8. In the year 2000 a shop sold 150 computers. Each year the shop sold 10 more computers than the year before, so that the shop sold 160 computers in 2001, 170 computers in 2002, and so on forming an arithmetic sequence.

1. Show that the shop sold 220 computers in 2007.
2. Calculate the total number of computers the shop sold from 2000 to 2013 inclusive. In the year 2000, the selling price of each computer was \(\pounds 900\). The selling price fell by \(\pounds 20\) each year, so that in 2001 the selling price was \(\pounds 880\), in 2002 the selling price was \(\pounds 860\), and so on forming an arithmetic sequence.
3. In a particular year, the selling price of each computer in \(\pounds s\) was equal to three times the number of computers the shop sold in that year. By forming and solving an equation, find the year in which this occurred.

4 Sam starts a job with an annual salary of \(\pounds 16000\). It is promised that the salary will go up by the same amount every year. In the second year Sam is paid \(\pounds 17200\).

1. Find Sam's salary in the tenth year.
2. Find the number of complete years needed for Sam's total salary to first exceed \(\pounds 500000\).
3. Comment on how realistic this model may be in the long term.

1. There were 2100 tonnes of wheat harvested on a farm during 2017.

The mass of wheat harvested during each subsequent year is expected to increase by \(1.2 \%\) per year.

1. Find the total mass of wheat expected to be harvested from 2017 to 2030 inclusive, giving your answer to 3 significant figures. Each year it costs
   • £5.15 per tonne to harvest the first 2000 tonnes of wheat
   • £6.45 per tonne to harvest wheat in excess of 2000 tonnes
   • Use this information to find the expected cost of harvesting the wheat from 2017 to 2030 inclusive. Give your answer to the nearest \(\pounds 1000\)

6. (a) An arithmetic series has first term \(a\) and common difference \(d\). Prove that the sum of the first \(n\) terms of the series is \(\frac { 1 } { 2 } n [ 2 a + ( n - 1 ) d ]\). A company made a profit of \(\pounds 54000\) in the year 2001. A model for future performance assumes that yearly profits will increase in an arithmetic sequence with common difference \(\pounds d\). This model predicts total profits of \(\pounds 619200\) for the 9 years 2001 to 2009 inclusive.
(b) Find the value of \(d\). Using your value of \(d\),
(c) find the predicted profit for the year 2011.

8. Amy plans to join a savings scheme in which she will pay in \(\pounds 500\) at the start of each year. One scheme that she is considering pays 6\% interest on the amount in the account at the end of each year. For this scheme,

1. find the amount of interest paid into the account at the end of the second year,
2. show that after interest is paid at the end of the eighth year, the amount in the account will be \(\pounds 5246\) to the nearest pound. Another scheme that she is considering pays \(0.5 \%\) interest on the amount in the account at the end of each month.
3. Find, to the nearest pound, how much more or less will be in the account at the end of the eighth year under this scheme.

2

1. Ben saves his pocket money as follows.
   Each week he puts money into his piggy bank (which pays no interest). In the first week he puts in 10p. In the second week he puts in 12p. In the third week he puts in 14p, and so on. How much money does Ben have in his piggy bank after 25 weeks?
2. On January 1st Shirley invests \(\pounds 500\) in a savings account that pays compound interest at \(3 \%\) per annum. She makes no further payments into this account. The interest is added on 31st December each year.
   1. Find the number of years after which her investment will first be worth more than \(\pounds 600\).
   2. State an assumption that you have made in answering part (ii)(a).

a) The \(3 ^ { \text {rd } } , 19 ^ { \text {th } }\) and \(67 ^ { \text {th } }\) terms of an arithmetic sequence form a geometric sequence. Given that the arithmetic sequence is increasing and that the first term is 3 , find the common difference of the arithmetic sequence. b) A firm has 100 employees on a particular Monday. The next day it adds 12 employees onto its staff and continues to do so on every successive working day, from Monday to Friday.
i) Find the number of employees at the end of the \(8 { } ^ { \text {th } }\) week.
ii) Each employee is paid \(\pounds 55\) per working day. Determine the total wage bill for the 8 week period.