Real-world arithmetic sequence application

4. A company, which is making 140 bicycles each week, plans to increase its production. The number of bicycles produced is to be increased by \(d\) each week, starting from 140 in week 1 , to \(140 + d\) in week 2 , to \(140 + 2 d\) in week 3 and so on, until the company is producing 206 in week 12.

1. Find the value of \(d\). After week 12 the company plans to continue making 206 bicycles each week.
2. Find the total number of bicycles that would be made in the first 52 weeks starting from and including week 1.

10. In this question you must show detailed reasoning. Owen wants to train for 12 weeks in preparation for running a marathon. During the 12-week period he will run every Sunday and every Wednesday.

• On Sunday in week 1 he will run 15 km
• On Sunday in week 12 he will run 37 km

He considers two different 12-week training plans. In training plan \(A\), he will increase the distance he runs each Sunday by the same amount.

1. Calculate the distance he will run on Sunday in week 5 under training plan \(A\). In training plan \(B\), he will increase the distance he runs each Sunday by the same percentage.
2. Calculate the distance he will run on Sunday in week 5 under training plan \(B\). Give your answer in km to one decimal place. Owen will also run a fixed distance, \(x \mathrm {~km}\), each Wednesday over the 12-week period. Given that
   • \(x\) is an integer
   • the total distance that Owen will run on Sundays and Wednesdays over the 12 weeks will not exceed 360 km
   • 1. find the maximum value of \(x\), if he uses training plan \(A\),
     2. find the maximum value of \(x\), if he uses training plan \(B\).
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8. A metal post is repeatedly hit in order to drive it into the ground. Given that

• on the 1st hit, the post is driven 100 mm into the ground
• on the 2nd hit, the post is driven an additional 98 mm into the ground
• on the 3rd hit, the post is driven an additional 96 mm into the ground
• the additional distances the post travels on each subsequent hit form an arithmetic sequence
  1. show that the post is driven an additional 62 mm into the ground with the 20th hit.
  2. Find the total distance that the post has been driven into the ground after 20 hits.

Given that for each subsequent hit after the 20th hit

• the additional distances the post travels form a geometric sequence with common ratio \(r\)
• on the 22 nd hit, the post is driven an additional 60 mm into the ground
• find the value of \(r\), giving your answer to 3 decimal places.

After a total of \(N\) hits, the post will have been driven more than 3 m into the ground.

Find, showing all steps in your working, the smallest possible value of \(N\).

1. Adina is saving money to buy a new computer. She saves \(\pounds 5\) in week \(1 , \pounds 5.25\) in week 2 , \(\pounds 5.50\) in week 3 and so on until she has enough money, in total, to buy the computer.

She decides to model her savings using either an arithmetic series or a geometric series.
Using the information given,

1. 1. state with a reason whether an arithmetic series or a geometric series should be used,
   2. write down an expression, in terms of \(n\), for the amount, in pounds ( \(\pounds\) ), saved in week \(n\). Given that the computer Adina wants to buy costs \(\pounds 350\)
2. find the number of weeks it will take for Adina to save enough money to buy the computer.
   

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6

1. John aims to pay a certain amount of money each month into a pension fund. He plans to pay \(\pounds 100\) in the first month, and then to increase the amount paid by \(\pounds 5\) each month, i.e. paying \(\pounds 105\) in the second month, \(\pounds 110\) in the third month, etc. If John continues making payments according to this plan for 240 months, calculate
   (a) how much he will pay in the final month,
   (b) how much he will pay altogether over the whole period.
2. Rachel also plans to pay money monthly into a pension fund over a period of 240 months, starting with \(\pounds 100\) in the first month. Her monthly payments will form a geometric progression, and she will pay \(\pounds 1500\) in the final month. Calculate how much Rachel will pay altogether over the whole period.

11

1. In a 'Make Ten' quiz game, contestants get \(\pounds 10\) for answering the first question correctly, then a further \(\pounds 20\) for the second question, then a further \(\pounds 30\) for the third, and so on, until they get a question wrong and are out of the game.
   (A) Haroon answers six questions correctly. Show that he receives a total of \(\pounds 210\).
   (B) State, in a simple form, a formula for the total amount received by a contestant who answers \(n\) questions correctly. Hence find the value of \(n\) for a contestant who receives \(\pounds 10350\) from this game.
2. In a 'Double Your Money' quiz game, contestants get \(\pounds 5\) for answering the first question correctly, then a further \(\pounds 10\) for the second question, then a further \(\pounds 20\) for the third, and so on doubling the amount for each question until they get a question wrong and are out of the game.
   (A) Gary received \(\pounds 75\) from the game. How many questions did he get right?
   (B) Bethan answered 9 questions correctly. How much did she receive from the game?
   (C) State a formula for the total amount received by a contestant who answers \(n\) questions correctly. Hence find the value of \(n\) for a contestant in this game who receives \(\pounds 2621435\).

3 On his \(1 ^ { \text {st } }\) birthday, John was given \(\pounds 5\) by his Uncle Fred. On each succeeding birthday, Uncle Fred gave a sum of money that was \(\pounds 3\) more than the amount he gave on the last birthday.

1. How much did Uncle Fred give John on his \(8 { } ^ { \text {th } }\) birthday?
2. On what birthday did the gift from Uncle Fred result in the total sum given on all birthdays exceeding £200?

8. A store begins to stock a new range of DVD players and achieves sales of \(\pounds 1500\) of these products during the first month. In a model it is assumed that sales will decrease by \(\pounds x\) in each subsequent month, forming an arithmetic sequence. Given that sales total \(\pounds 8100\) during the first six months, use the model to

1. find the value of \(x\),
2. find the expected value of sales in the eighth month,
3. show that the expected total of sales in pounds during the first \(n\) months is given by \(k n ( 51 - n )\), where \(k\) is an integer to be found.
4. Explain why this model cannot be valid over a long period of time.

1

1. In a 'Make Ten' quiz game, contestants get \(\pounds 10\) for answering the first question correctly, then a further \(\pounds 20\) for the second question, then a further \(\pounds 30\) for the third, and so on, until they get a question wrong and are out of the game.
   (A) Haroon answers six questions correctly. Show that he receives a total of \(\pounds 210\).
   (B) State, in a simple form, a formula for the total amount received by a contestant who answers \(n\) questions correctly. Hence find the value of \(n\) for a contestant who receives \(\pounds 10350\) from this game.
2. In a 'Double Your Money' quiz game, contestants get \(\pounds 5\) for answering the first question correctly, then a further \(\pounds 10\) for the second question, then a further \(\pounds 20\) for the third, and so on doubling the amount for each question until they get a question wrong and are out of the game.
   (A) Gary received \(\pounds 75\) from the game. How many questions did he get right?
   (B) Bethan answered 9 questions correctly. How much did she receive from the game?
   (C) State a formula for the total amount received by a contestant who answers \(n\) questions correctly. Hence find the value of \(n\) for a contestant in this game who receives \(\pounds 2621435\).

4

1. André is playing a game where he makes piles of counters. He puts 3 counters in the first pile. Each successive pile he makes has 2 more counters in it than the previous one.
   1. How many counters are there in his sixth pile?
   2. André makes ten piles of counters. How many counters has he used altogether?
2. In another game, played with an ordinary fair die and counters, Betty needs to throw a six to start. The probability \(\mathrm { P } _ { n }\) of Betty starting on her \(n\)th throw is given by $$P _ { n } = \frac { 1 } { 6 } \times \left( \frac { 5 } { 6 } \right) ^ { n - 1 }$$
   1. Calculate \(\mathrm { P } _ { 4 }\). Give your answer as a fraction.
   2. The values \(\mathrm { P } _ { 1 } , \mathrm { P } _ { 2 } , \mathrm { P } _ { 3 } , \ldots\) form an infinite geometric progression. State the first term and the common ratio of this progression. Hence show that \(\mathrm { P } _ { 1 } + \mathrm { P } _ { 2 } + \mathrm { P } _ { 3 } + \ldots = 1\).
   3. Given that \(\mathrm { P } _ { n } < 0.001\), show that \(n\) satisfies the inequality $$n > \frac { \log _ { 10 } 0.006 } { \log _ { 10 } \left( \frac { 5 } { 6 } \right) } + 1$$ Hence find the least value of \(n\) for which \(\mathrm { P } _ { n } < 0.001\).

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.

6 Sarah is carrying out a series of experiments which involve using increasing amounts of a chemical. In the first experiment she uses 6 g of the chemical and in the second experiment she uses 7.8 g of the chemical.

1. Given that the amounts of the chemical used form an arithmetic progression, find the total amount of chemical used in the first 30 experiments.
2. Instead it is given that the amounts of the chemical used form a geometric progression. Sarah has a total of 1800 g of the chemical available. Show that \(N\), the greatest number of experiments possible, satisfies the inequality $$1.3 ^ { N } \leqslant 91 ,$$ and use logarithms to calculate the value of \(N\).

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.

7 Two students, Anna and Ben, are starting a revision programme. They will both revise for 30 minutes on Day 1. Anna will increase her revision time by 15 minutes for every subsequent day. Ben will increase his revision time by \(10 \%\) for every subsequent day.

1. Verify that on Day 10 Anna does 94 minutes more revision than Ben, correct to the nearest minute. Let Day \(X\) be the first day on which Ben does more revision than Anna.
2. Show that \(X\) satisfies the inequality \(X > \log _ { 1.1 } ( 0.5 X + 0.5 ) + 1\).
3. Use the iterative formula \(x _ { n + 1 } = \log _ { 1.1 } \left( 0.5 x _ { n } + 0.5 \right) + 1\) with \(x _ { 1 } = 10\) to find the value of \(X\). You should show the result of each iteration.
4. 1. Give a reason why Anna's revision programme may not be realistic.
   2. Give a different reason why Ben's revision programme may not be realistic.

1. Ben starts a new company.

• In year 1 his profits will be \(\pounds 24000\).
• In year 11 his profit is predicted to be \(\pounds 64000\).

Model \(\boldsymbol { P }\) assumes that his profit will increase by the same amount each year.
a. According to model \(\boldsymbol { P }\), determine Ben's profit in year 5. Model \(\boldsymbol { Q }\) assumes that his profit will increase by the same percentage each year.
b. According to model \(\boldsymbol { Q }\), determine Ben's profit in year 5 . Give your answer to the nearest £10.

5. A scientist is studying a population of lizards on an island and uses the linear model \(P = a + b t\) to predict the future population of the lizard where \(P\) is the population and \(t\) is the time in years after the start of the study. Given that

• The number of population was 900 , exactly 5 years after the start of the study.
• The number of population was 1200 , exactly 8 years after the start of the study.
  a. find a complete equation for the model.
  b. Sketch the graph of the population for the first 10 years.
  c. Suggest one criticism of this model.

1. A car has six forward gears.

The fastest speed of the car

• in \(1 ^ { \text {st } }\) gear is \(28 \mathrm {~km} \mathrm {~h} ^ { - 1 }\)
• in \(6 ^ { \text {th } }\) gear is \(115 \mathrm {~km} \mathrm {~h} ^ { - 1 }\)

Given that the fastest speed of the car in successive gears is modelled by an arithmetic sequence,

1. find the fastest speed of the car in \(3 { } ^ { \text {rd } }\) gear. Given that the fastest speed of the car in successive gears is modelled by a geometric sequence,
2. find the fastest speed of the car in \(5 ^ { \text {th } }\) gear.

3 Sam invested in a shares scheme. The value, \(\pounds V\), of Sam's shares was reported \(t\) months after investment.

• Exactly 6 months after investment, the value of Sam's shares was \(\pounds 2375\).
• Exactly 1 year after investment, the value of Sam's shares was \(\pounds 2825\).
  1. Using a straight-line model, determine an equation for \(V\) in terms of \(t\).

Sam's original investment in the scheme was \(\pounds 1900\).

Explain whether or not this fact supports the use of the straight-line model in part (a).

6 Aleela and Baraka are saving to buy a car. Aleela saves \(\pounds 50\) in the first month. She increases the amount she saves by \(\pounds 20\) each month.

1. Calculate how much she saves in two years. Baraka also saves \(\pounds 50\) in the first month. The amount he saves each month is \(12 \%\) more than the amount he saved in the previous month.
2. Explain why the amounts Baraka saves each month form a geometric sequence.
3. Determine whether Baraka saves more in two years than Aleela. Answer all the questions
   Section B (77 marks)

4. In the first month after opening, a mobile phone shop sold 280 phones. A model for future trading assumes that sales will increase by \(x\) phones per month for the next 35 months, so that \(( 280 + x )\) phones will be sold in the second month, \(( 280 + 2 x )\) in the third month, and so on. Using this model with \(x = 5\), calculate

1. 1. the number of phones sold in the 36th month,
   2. the total number of phones sold over the 36 months. The shop sets a sales target of 17000 phones to be sold over the 36 months.
      Using the same model,
2. find the least value of \(x\) required to achieve this target.

6. Each year, for 40 years, Anne will pay money into a savings scheme. In the first year she pays \(\pounds 500\). Her payments then increase by \(\pounds 50\) each year, so that she pays \(\pounds 550\) in the second year, \(\pounds 600\) in the third year, and so on.

1. Find the amount that Anne will pay in the 40th year.
2. Find the total amount that Anne will pay in over the 40 years. Over the same 40 years, Brian will also pay money into the savings scheme. In the first year he pays in \(\pounds 890\) and his payments then increase by \(\pounds d\) each year. Given that Brian and Anne will pay in exactly the same amount over the 40 years,
3. find the value of \(d\).

7. Each year, for 40 years, Anne will pay money into a savings scheme. In the first year she pays \(\pounds 500\). Her payments then increase by \(\pounds 50\) each year, so that she pays \(\pounds 550\) in the second year, \(\pounds 600\) in the third year, and so on.

1. Find the amount that Anne will pay in the 40th year.
2. Find the total amount that Anne will pay in over the 40 years. Over the same 40 years, Brian will also pay money into the savings scheme. In the first year he pays in \(\pounds 890\) and his payments then increase by \(\pounds d\) each year. Given that Brian and Anne will pay in exactly the same amount over the 40 years,
3. find the value of \(d\).