1.04h Arithmetic sequences: nth term and sum formulae

7. An athlete prepares for a race by completing a practice run on each of 11 consecutive days. On each day after the first day, he runs further than he ran on the previous day. The lengths of his 11 practice runs form an arithmetic sequence with first term \(a \mathrm {~km}\) and common difference \(d \mathrm {~km}\). He runs 9 km on the 11th day, and he runs a total of 77 km over the 11 day period.
Find the value of \(a\) and the value of \(d\).

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 \(d\).

5. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by $$\begin{aligned} a _ { 1 } & = k \\ a _ { n + 1 } & = 5 a _ { n } + 3 , \quad n \geqslant 1 , \end{aligned}$$ where \(k\) is a positive integer.

1. Write down an expression for \(a _ { 2 }\) in terms of \(k\).
2. Show that \(a _ { 3 } = 25 k + 18\).
3. 1. Find \(\sum _ { r = 1 } ^ { 4 } a _ { r }\) in terms of \(k\), in its simplest form.
   2. Show that \(\sum _ { r = 1 } ^ { 4 } a _ { r }\) is divisible by 6 .

1. (a) Calculate the sum of all the even numbers from 2 to 100 inclusive,

$$2 + 4 + 6 + \ldots \ldots + 100$$ (b) In the arithmetic series $$k + 2 k + 3 k + \ldots \ldots + 100$$ \(k\) is a positive integer and \(k\) is a factor of 100 .

1. Find, in terms of \(k\), an expression for the number of terms in this series.
2. Show that the sum of this series is $$50 + \frac { 5000 } { k }$$ (c) Find, in terms of \(k\), the 50th term of the arithmetic sequence $$( 2 k + 1 ) , ( 4 k + 4 ) , ( 6 k + 7 ) , \ldots \ldots ,$$ giving your answer in its simplest form.

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.

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.

9. An arithmetic series has first term \(a\) and common difference \(d\).

1. Prove that the sum of the first \(n\) terms of the series is \(\frac { 1 } { 2 } n [ 2 a + ( n - 1 ) d ] .\) (4)
   A polygon has 16 sides. The lengths of the sides of the polygon, starting with the shortest side, form an arithmetic sequence with common difference \(d \mathrm {~cm}\).
   The longest side of the polygon has length 6 cm and the perimeter of the polygon is 72 cm .
   Find
2. the length of the shortest side of the polygon,
   (5)
3. the value of \(d\).
   (2) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \end{tabular} & Leave blank
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8. (i) An arithmetic series has first term \(a\) and common difference \(d\). Prove that the sum to \(n\) terms of this series is $$\frac { n } { 2 } \{ 2 a + ( n - 1 ) d \}$$ (ii) A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is given by $$u _ { n } = 5 n + 3 ( - 1 ) ^ { n }$$ Find the value of

1. \(u _ { 5 }\)
2. \(\sum _ { n = 1 } ^ { 59 } u _ { n }\)

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

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. Wheat is grown on a farm.

• In year 1 , the farm produced 300 tonnes of wheat.
• In year 12 , the farm is predicted to produce 4000 tonnes of wheat.

Model \(A\) assumes that the amount of wheat produced on the farm will increase by the same amount each year.

1. Using model \(A\), find the amount of wheat produced on the farm in year 4. Give your answer to the nearest 10 tonnes. Model \(B\) assumes that the amount of wheat produced on the farm will increase by the same percentage each year.
2. Using model \(B\), find the amount of wheat produced on the farm in year 2. Give your answer to the nearest 10 tonnes.
3. Calculate, according to the two models, the difference between the total amounts of wheat predicted to be produced on the farm from year 1 to year 12 inclusive. Give your answer to the nearest 10 tonnes.

7. Kim starts working for a company.

• In year 1 her annual salary will be \(\pounds 16200\)
• In year 10 her annual salary is predicted to be \(\pounds 31500\)

Model \(A\) assumes that her annual salary will increase by the same amount each year.

1. According to model \(A\), determine Kim's annual salary in year 2 . Model \(B\) assumes that her annual salary will increase by the same percentage each year.
2. According to model \(B\), determine Kim's annual salary in year 2 . Give your answer to the nearest \(\pounds 10\)
3. Calculate, according to the two models, the difference between the total amounts that Kim is predicted to earn from year 1 to year 10 inclusive. Give your answer to the nearest £10

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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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 $$S _ { n } > 8000$$

9. Solutions based entirely on graphical or numerical methods are not acceptable in this question.

1. Solve, for \(0 \leqslant \theta < 180 ^ { \circ }\), the equation $$3 \sin \left( 2 \theta - 10 ^ { \circ } \right) = 1$$ giving your answers to one decimal place.
2. The first three terms of an arithmetic sequence are $$\sin \alpha , \frac { 1 } { \tan \alpha } \text { and } 2 \sin \alpha$$ where \(\alpha\) is a constant.
   1. Show that \(2 \cos \alpha = 3 \sin ^ { 2 } \alpha\) Given that \(\pi < \alpha < 2 \pi\),
   2. find, showing all working, the value of \(\alpha\) to 3 decimal places.

5. Ben is saving for the deposit for a house over a period of 60 months. Ben saves \(\pounds 100\) in the first month and in each subsequent month, he saves \(\pounds 5\) more than the previous month, so that he saves \(\pounds 105\) in the second month, \(\pounds 110\) in the third month, and so on, forming an arithmetic sequence.

1. Find the amount Ben saves in the 40th month.
2. Find the total amount Ben saves over the 60 -month period. Lina is also saving for a deposit for a house.
   Lina saves \(\pounds 600\) in the first month and in each subsequent month, she saves \(\pounds 10\) less than the previous month, so that she saves \(\pounds 590\) in the second month, \(\pounds 580\) in the third month, and so on, forming an arithmetic sequence. Given that, after \(n\) months, Lina will have saved exactly \(\pounds 18200\) for her deposit,
3. form an equation in \(n\) and show that it can be written as $$n ^ { 2 } - 121 n + 3640 = 0$$
4. Solve the equation in part (c).
5. State, with a reason, which of the solutions to the equation in part (c) is not a sensible value for \(n\).