Real-world arithmetic sequence application

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.

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.
   (a) Find the number of years after which her investment will first be worth more than \(\pounds 600\).
   (b) State an assumption that you have made in answering part (ii)(a).

1. (a) Given that \(8 = 2 ^ { k }\), write down the value of \(k\).
   (b) Given that \(4 ^ { x } = 8 ^ { 2 - x }\), find the value of \(x\).
2. The equation \(x ^ { 2 } + 5 k x + 2 k = 0\), where \(k\) is a constant, has real roots.
   (a) Prove that \(k ( 25 k - 8 ) \geq 0\).
   (b) Hence find the set of possible values of \(k\).
   (c) Write down the values of \(k\) for which the equation \(x ^ { 2 } + 5 k x + 2 k = 0\) has equal roots.
3. 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.
   (a) Find the amount that Anne will pay in the 40th year.
   (b) 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,
(c) find the value of \(d\).

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. 1. 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
3. (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.