Real-world arithmetic sequence application

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.

10
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1. Find angle \(A B C\). The point \(D\) is such that \(A B C D\) is a parallelogram.
2. Find the position vector of \(D\). 9 The function f is such that \(\mathrm { f } ( x ) = 3 - 4 \cos ^ { k } x\), for \(0 \leqslant x \leqslant \pi\), where \(k\) is a constant.
3. In the case where \(k = 2\),
   (a) find the range of f,
   (b) find the exact solutions of the equation \(\mathrm { f } ( x ) = 1\).
4. In the case where \(k = 1\),
   (a) sketch the graph of \(y = \mathrm { f } ( x )\),
   (b) state, with a reason, whether f has an inverse. 10 (a) A circle is divided into 6 sectors in such a way that the angles of the sectors are in arithmetic progression. The angle of the largest sector is 4 times the angle of the smallest sector. Given that the radius of the circle is 5 cm , find the perimeter of the smallest sector.
   (b) The first, second and third terms of a geometric progression are \(2 k + 3 , k + 6\) and \(k\), respectively. Given that all the terms of the geometric progression are positive, calculate
5. the value of the constant \(k\),
6. the sum to infinity of the progression.

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.
   (a) How many litres will be lost on the 30th day after filling?
   (b) The tank becomes empty during the \(n\)th day after filling. Find the value of \(n\).
2. 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.]

5 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.
2. He reaches his 13 kg target during week \(n\). Use your answer to part (i) to find the value of \(n\).
   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.
3. Calculate his total weight loss after 20 weeks and show that he can never reach his target.

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 .

10

1. An arithmetic progression contains 25 terms and the first term is - 15 . The sum of all the terms in the progression is 525. Calculate
   1. the common difference of the progression,
   2. the last term in the progression,
   3. the sum of all the positive terms in the progression.
2. A college agrees a sponsorship deal in which grants will be received each year for sports equipment. This grant will be \(\\) 4000\( in 2012 and will increase by \)5 \%$ each year. Calculate
   1. the value of the grant in 2022,
   2. the total amount the college will receive in the years 2012 to 2022 inclusive.

7

1. An athlete runs the first mile of a marathon in 5 minutes. His speed reduces in such a way that each mile takes 12 seconds longer than the preceding mile.
   1. Given that the \(n\)th mile takes 9 minutes, find the value of \(n\).
   2. Assuming that the length of the marathon is 26 miles, find the total time, in hours and minutes, to complete the marathon.
2. The second and third terms of a geometric progression are 48 and 32 respectively. Find the sum to infinity of the progression.

8. A 25-year programme for building new houses began in Core Town in the year 1986 and finished in the year 2010. The number of houses built each year form an arithmetic sequence. Given that 238 houses were built in the year 2000 and 108 were built in the year 2010, find

1. the number of houses built in 1986, the first year of the building programme,
2. the total number of houses built in the 25 years of the programme.

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.

16. Maria trains for a triathlon, which involves swimming, cycling and running. On the first day of training she swims 1.5 km and then she swims 1.5 km on each of the following days.

1. Find the total distance that Maria swims in the first 17 days of training. Maria also runs 1.5 km on the first day of training and on each of the following days she runs 0.25 km further than on the previous day. So she runs 1.75 km on the second day and 2 km on the third day and so on.
2. Find how far Maria runs on the 17th day of training. Maria also cycles 1.5 km on the first day of training and on each of the following days she cycles \(5 \%\) further than on the previous day.
3. Find the total distance that Maria cycles in the first 17 days of training.
4. Find the total distance Maria travels by swimming, running and cycling in the first 17 days of training. Maria needs to cycle 40 km in the triathlon.
5. On which day of training does Maria first cycle more than 40 km ?

7. Jill gave money to a charity over a 20 -year period, from Year 1 to Year 20 inclusive. She gave \(\pounds 150\) in Year \(1 , \pounds 160\) in Year 2, \(\pounds 170\) in Year 3, and so on, so that the amounts of money she gave each year formed an arithmetic sequence.

1. Find the amount of money she gave in Year 10.
2. Calculate the total amount of money she gave over the 20 -year period. Kevin also gave money to the charity over the same 20 -year period. He gave \(\pounds A\) in Year 1 and the amounts of money he gave each year increased, forming an arithmetic sequence with common difference \(\pounds 30\). The total amount of money that Kevin gave over the 20 -year period was twice the total amount of money that Jill gave.
3. Calculate the value of \(A\).

1. A company offers two salary schemes for a 10 -year period, Year 1 to Year 10 inclusive.

Scheme 1: Salary in Year 1 is \(\pounds P\).
Salary increases by \(\pounds ( 2 T )\) each year, forming an arithmetic sequence.
Scheme 2: Salary in Year 1 is \(\pounds ( P + 1800 )\).
Salary increases by \(\pounds T\) each year, forming an arithmetic sequence.

1. Show that the total earned under Salary Scheme 1 for the 10-year period is $$\pounds ( 10 P + 90 T )$$ For the 10-year period, the total earned is the same for both salary schemes.
2. Find the value of \(T\). For this value of \(T\), the salary in Year 10 under Salary Scheme 2 is \(\pounds 29850\)
3. Find the value of \(P\).

1. Lewis played a game of space invaders. He scored points for each spaceship that he captured.

Lewis scored 140 points for capturing his first spaceship.
He scored 160 points for capturing his second spaceship, 180 points for capturing his third spaceship, and so on. The number of points scored for capturing each successive spaceship formed an arithmetic sequence.

1. Find the number of points that Lewis scored for capturing his 20th spaceship.
2. Find the total number of points Lewis scored for capturing his first 20 spaceships. Sian played an adventure game. She scored points for each dragon that she captured. The number of points that Sian scored for capturing each successive dragon formed an arithmetic sequence. Sian captured \(n\) dragons and the total number of points that she scored for capturing all \(n\) dragons was 8500 . Given that Sian scored 300 points for capturing her first dragon and then 700 points for capturing her \(n\)th dragon,
3. find the value of \(n\).

7. Sue is training for a marathon. Her training includes a run every Saturday starting with a run of 5 km on the first Saturday. Each Saturday she increases the length of her run from the previous Saturday by 2 km .

1. Show that on the 4th Saturday of training she runs 11 km .
2. Find an expression, in terms of \(n\), for the length of her training run on the \(n\)th Saturday.
3. Show that the total distance she runs on Saturdays in \(n\) weeks of training is \(n ( n + 4 ) \mathrm { km }\). On the \(n\)th Saturday Sue runs 43 km .
4. Find the value of \(n\).
5. Find the total distance, in km , Sue runs on Saturdays in \(n\) weeks of training.

5. A 40-year building programme for new houses began in Oldtown in the year 1951 (Year 1) and finished in 1990 (Year 40). The numbers of houses built each year form an arithmetic sequence with first term \(a\) and common difference \(d\). Given that 2400 new houses were built in 1960 and 600 new houses were built in 1990, find

1. the value of \(d\),
2. the value of \(a\),
3. the total number of houses built in Oldtown over the 40-year period.

1. A farmer has a pay scheme to keep fruit pickers working throughout the 30 day season. He pays \(\pounds a\) for their first day, \(\pounds ( a + d )\) for their second day, \(\pounds ( a + 2 d )\) for their third day, and so on, thus increasing the daily payment by \(\pounds d\) for each extra day they work.

A picker who works for all 30 days will earn \(\pounds 40.75\) on the final day.

1. Use this information to form an equation in \(a\) and \(d\). A picker who works for all 30 days will earn a total of \(\pounds 1005\)
2. Show that \(15 ( a + 40.75 ) = 1005\)
3. Hence find the value of \(a\) and the value of \(d\).

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. Each year, Abbie pays into a savings scheme. In the first year she pays in \(\pounds 500\). Her payments then increase by \(\pounds 200\) each year so that she pays \(\pounds 700\) in the second year, \(\pounds 900\) in the third year and so on.

1. Find out how much Abbie pays into the savings scheme in the tenth year. Abbie pays into the scheme for \(n\) years until she has paid in a total of \(\pounds 67200\).
2. Show that \(n ^ { 2 } + 4 n - 24 \times 28 = 0\)
3. Hence find the number of years that Abbie pays into the savings scheme.

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. Xin has been given a 14 day training schedule by her coach.

Xin will run for \(A\) minutes on day 1 , where \(A\) is a constant.
She will then increase her running time by ( \(d + 1\) ) minutes each day, where \(d\) is a constant.

1. Show that on day 14 , Xin will run for $$( A + 13 d + 13 ) \text { minutes. }$$ Yi has also been given a 14 day training schedule by her coach.
   Yi will run for \(( A - 13 )\) minutes on day 1 .
   She will then increase her running time by ( \(2 d - 1\) ) minutes each day.
   Given that Yi and Xin will run for the same length of time on day 14,
2. find the value of \(d\). Given that Xin runs for a total time of 784 minutes over the 14 days,
3. find the value of \(A\).

Jess started work 20 years ago. In year 1 her annual salary was \(\pounds 17000\). Her annual salary increased by \(\pounds 1500\) each year, so that her annual salary in year 2 was \(\pounds 18500\), in year 3 it was \(\pounds 20000\) and so on, forming an arithmetic sequence. This continued until she reached her maximum annual salary of \(\pounds 32000\) in year \(k\). Her annual salary then remained at \(\pounds 32000\).

1. Find the value of the constant \(k\).
2. Calculate the total amount that Jess has earned in the 20 years.

9. On John's 10th birthday he received the first of an annual birthday gift of money from his uncle. This first gift was \(\pounds 60\) and on each subsequent birthday the gift was \(\pounds 15\) more than the year before. The amounts of these gifts form an arithmetic sequence.

1. Show that, immediately after his 12th birthday, the total of these gifts was \(\pounds 225\)
2. Find the amount that John received from his uncle as a birthday gift on his 18th birthday.
3. Find the total of these birthday gifts that John had received from his uncle up to and including his 21st birthday. When John had received \(n\) of these birthday gifts, the total money that he had received from these gifts was \(\pounds 3375\)
4. Show that \(n ^ { 2 } + 7 n = 25 \times 18\)
5. Find the value of \(n\), when he had received \(\pounds 3375\) in total, and so determine John's age at this time.