Real-world AP: find n satisfying a condition

9 A water tank holds 2000 litres when full. A small hole in the base is gradually getting bigger so that each day a greater amount of water is lost.

1. On the first day after filling, 10 litres of water are lost and this increases by 2 litres each day.
   1. How many litres will be lost on the 30th day after filling?
   2. The tank becomes empty during the \(n\)th day after filling. Find the value of \(n\).
   3. Assume instead that 10 litres of water are lost on the first day and that the amount of water lost increases by \(10 \%\) on each succeeding day. Find what percentage of the original 2000 litres is left in the tank at the end of the 30th day after filling.
      [0pt] [Questions 10 and 11 are printed on the next page.]

3

1. A debt of \(\\) 3726\( is repaid by weekly payments which are in arithmetic progression. The first payment is \)\\( 60\) and the debt is fully repaid after 48 weeks. Find the third payment.
2. Find the sum to infinity of the geometric progression whose first term is 6 and whose second term is 4 .

9. A car manufacturer currently makes 1000 cars each week. The manufacturer plans to increase the number of cars it makes each week. The number of cars made will be increased by 20 each week from 1000 in week 1, to 1020 in week 2, to 1040 in week 3 and so on, until 1500 cars are made in week \(N\).

1. Find the value of \(N\). The car manufacturer then plans to continue to make 1500 cars each week.
2. Find the total number of cars that will be made in the first 50 weeks starting from and including week 1.

6. A boy saves some money over a period of 60 weeks. He saves 10 p in week 1 , 15 p in week \(2,20 \mathrm { p }\) in week 3 and so on until week 60 . His weekly savings form an arithmetic sequence.

1. Find how much he saves in week 15
2. Calculate the total amount he saves over the 60 week period. The boy's sister also saves some money each week over a period of \(m\) weeks. She saves 10 p in week \(1,20 \mathrm { p }\) in week \(2,30 \mathrm { p }\) in week 3 and so on so that her weekly savings form an arithmetic sequence. She saves a total of \(\pounds 63\) in the \(m\) weeks.
3. Show that $$m ( m + 1 ) = 35 \times 36$$
4. Hence write down the value of \(m\).

7. A company, which is making 200 mobile phones each week, plans to increase its production. The number of mobile phones produced is to be increased by 20 each week from 200 in week 1 to 220 in week 2, to 240 in week 3 and so on, until it is producing 600 in week \(N\).

1. Find the value of \(N\). The company then plans to continue to make 600 mobile phones each week.
2. Find the total number of mobile phones that will be made in the first 52 weeks starting from and including week 1.

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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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 \(\pounds 2621435\).

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 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. As part of a new training programme, Habib decides to do sit-ups every day. He plans to do 20 per day in the first week, 22 per day in the second week, 24 per day in the third week and so on, increasing the daily number of sit-ups by two at the start of each week.

1. Find the number of sit-ups that Habib will do in the fifth week.
2. Show that he will do a total of 1512 sit-ups during the first eight weeks. In the \(n\)th week of training, the number of sit-ups that Habib does is greater than 300 for the first time.
3. Find the value of \(n\).

5 An ice cream seller expects that the number of sales will increase by the same amount every week from May onwards. 150 ice creams are sold in Week 1 and 166 ice creams are sold in Week 2. The ice cream seller makes a profit of \(\pounds 1.25\) for each ice cream sold.

1. Find the expected profit in Week 10.
2. In which week will the total expected profits first exceed \(\pounds 5000\) ?
3. Give two reasons why this model may not be appropriate.

7 Business A made a \(\pounds 5000\) profit during its first year.
In each subsequent year, the profit increased by \(\pounds 1500\) so that the profit was \(\pounds 6500\) during the second year, \(\pounds 8000\) during the third year and so on. Business B made a \(\pounds 5000\) profit during its first year.
In each subsequent year, the profit was 90\% of the previous year's profit.

1. Find an expression for the total profit made by business A during the first \(n\) years. Give your answer in its simplest form.
2. Find an expression for the total profit made by business B during the first \(n\) years. Give your answer in its simplest form.
3. Find how many years it will take for the total profit of business A to reach \(\pounds 385000\).
4. Comment on the profits made by each business in the long term.

5 Ziad is training to become a long-distance swimmer. He trains every day by swimming lengths at his local pool.
The length of the pool is 25 metres.
Each day he increases the number of lengths that he swims by four.
On his first day of training, Ziad swims 10 lengths of the pool.
5

1. Write down an expression for the number of lengths Ziad will swim on his \(n\)th day of training. 5
2. (i) Ziad's target is to be able to swim at least 3000 metres in one day.
   Determine the minimum number of days he will need to train to reach his target.
   5 (b) (ii) Ziad's coach claims that when he reaches his target he will have covered a total distance of over 50000 metres. Determine if Ziad's coach is correct.

Two heavyweight boxers decide that they would be more successful if they competed in a lower weight class. For each boxer this would require a total weight loss of 13 kg. At the end of week 1 they have each recorded a weight loss of 1 kg and they both find that in each of the following weeks their weight loss is slightly less than the week before. Boxer A's weight loss in week 2 is 0.98 kg. It is given that his weekly weight loss follows an arithmetic progression.

1. Write down an expression for his total weight loss after \(x\) weeks. [1]
2. He reaches his 13 kg target during week \(n\). Use your answer to part (i) to find the value of \(n\). [2]

Boxer B's weight loss in week 2 is 0.92 kg and it is given that his weekly weight loss follows a geometric progression.

1. Calculate his total weight loss after 20 weeks and show that he can never reach his target. [4]

Ann has some sticks that are all of the same length. She arranges them in squares and has made the following 3 rows of patterns: Row 1 \(\square\) Row 2 \(\square\square\) Row 3 \(\square\square\square\) She notices that 4 sticks are required to make the single square in the first row, 7 sticks to make 2 squares in the second row and in the third row she needs 10 sticks to make 3 squares.

1. Find an expression, in terms of \(n\), for the number of sticks required to make a similar arrangement of \(n\) squares in the \(n\)th row. [3]

Ann continues to make squares following the same pattern. She makes 4 squares in the 4th row and so on until she has completed 10 rows.

1. Find the total number of sticks Ann uses in making these 10 rows. [3]

Ann started with 1750 sticks. Given that Ann continues the pattern to complete \(k\) rows but does not have sufficient sticks to complete the \((k + 1)\)th row,

1. show that \(k\) satisfies \((3k - 100)(k + 35) < 0\). [4]
2. Find the value of \(k\). [2]

Business A made a £5000 profit during its first year. In each subsequent year, the profit increased by £1500 so that the profit was £6500 during the second year, £8000 during the third year and so on. Business B made a £5000 profit during its first year. In each subsequent year, the profit was 90% of the previous year's profit.

1. Find an expression for the total profit made by business A during the first \(n\) years. Give your answer in its simplest form. [2]
2. Find an expression for the total profit made by business B during the first \(n\) years. Give your answer in its simplest form. [3]
3. Find how many years it will take for the total profit of business A to reach £385 000. [3]
4. Comment on the profits made by each business in the long term. [2]