1.04h Arithmetic sequences: nth term and sum formulae

10 In an arithmetic progression, the second term is 11 and the sum of the first 40 terms is 3030 . Find the first term and the common difference.

12 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?

2 Find the second and third terms in the sequence given by $$\begin{aligned} & u _ { 1 } = 5 \\ & u _ { n + 1 } = u _ { n } + 3 . \end{aligned}$$ Find also the sum of the first 50 terms of this sequence.

3 An arithmetic progression has tenth term 11.1 and fiftieth term 7.1. Find the first term and the common difference. Find also the sum of the first fifty terms of the progression.

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.

3 The 15th term of an arithmetic sequence is 88. The sum of the first 10 terms is 310 .
Determine the first term and the common difference.

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.

13. A construction company had a 30 -year programme to build new houses in Newtown. They began in the year 1991 (Year 1) and finished in 2020 (Year 30).
The company built 120 houses in year 1, 140 in year 2, 160 houses in year 3 and so on, so that the number of houses they built form an arithmetic sequence.
A total of 8400 new houses were built in year \(n\).
a. Show that $$n ^ { 2 } + 11 n - 840 = 0$$ b. Solve the equation $$n ^ { 2 } + 11 n - 840 = 0$$ and hence find in which year 8400 new houses were built.

4. (a) Show that \(\sum _ { r = 1 } ^ { 20 } \left( 2 ^ { r - 1 } - 3 - 4 r \right) = 1047675\) (b) A sequence has \(n\)th term \(u _ { n } = \sin \left( 90 n ^ { \circ } \right) n \geq 1\)

1. Find the order of the sequence.
2. Find \(\sum _ { r = 1 } ^ { 222 } u _ { r }\)

1. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 }\) is defined by

$$a _ { n } = \sin ^ { 2 } \left( \frac { n \pi } { 3 } \right)$$ Find the exact values of
a. i) \(a _ { 1 }\) ii) \(a _ { 2 }\) iii) \(a _ { 3 }\) b. Hence find the exact value of $$\sum _ { n = 1 } ^ { 100 } \left\{ n + \sin ^ { 2 } \left( \frac { n \pi } { 3 } \right) \right\}$$

1. A competitor is running a 20 kilometre race.

She runs each of the first 4 kilometres at a steady pace of 6 minutes per kilometre. After the first 4 kilometres, she begins to slow down. In order to estimate her finishing time, the time that she will take to complete each subsequent kilometre is modelled to be \(5 \%\) greater than the time that she took to complete the previous kilometre. Using the model,

1. show that her time to run the first 6 kilometres is estimated to be 36 minutes 55 seconds,
2. show that her estimated time, in minutes, to run the \(r\) th kilometre, for \(5 \leqslant r \leqslant 20\), is $$6 \times 1.05 ^ { r - 4 }$$
3. estimate the total time, in minutes and seconds, that she will take to complete the race.

1. 1. In an arithmetic series, the first term is \(a\) and the common difference is \(d\).

Show that $$S _ { n } = \frac { n } { 2 } [ 2 a + ( n - 1 ) d ]$$ (ii) James saves money over a number of weeks to buy a printer that costs \(\pounds 64\) He saves \(\pounds 10\) in week \(1 , \pounds 9.20\) in week \(2 , \pounds 8.40\) in week 3 and so on, so that the weekly amounts he saves form an arithmetic sequence. Given that James takes \(n\) weeks to save exactly \(\pounds 64\)

1. show that $$n ^ { 2 } - 26 n + 160 = 0$$
2. Solve the equation $$n ^ { 2 } - 26 n + 160 = 0$$
3. Hence state the number of weeks James takes to save enough money to buy the printer, giving a brief reason for your answer.

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.

1. (i) Show that \(\sum _ { r = 1 } ^ { 16 } \left( 3 + 5 r + 2 ^ { r } \right) = 131798\) (ii) A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by

$$u _ { n + 1 } = \frac { 1 } { u _ { n } } , \quad u _ { 1 } = \frac { 2 } { 3 }$$ Find the exact value of \(\sum _ { r = 1 } ^ { 100 } u _ { r }\)

1. (a) Express \(2 \cos \theta + 8 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\) Give the exact value of \(R\) and give the value of \(\alpha\) in radians to 3 decimal places.

The first three terms of an arithmetic sequence are $$\cos x \quad \cos x + \sin x \quad \cos x + 2 \sin x \quad x \neq n \pi$$ Given that \(S _ { 9 }\) represents the sum of the first 9 terms of this sequence as \(x\) varies,
(b) (i) find the exact maximum value of \(S _ { 9 }\) (ii) deduce the smallest positive value of \(x\) at which this maximum value of \(S _ { 9 }\) occurs.

1. Jamie takes out an interest-free loan of \(\pounds 8100\)

Jamie makes a payment every month to pay back the loan.
Jamie repays \(\pounds 400\) in month \(1 , \pounds 390\) in month \(2 , \pounds 380\) in month 3 , and so on, so that the amounts repaid each month form an arithmetic sequence.

1. Show that Jamie repays \(\pounds 290\) in month 12 After Jamie's \(N\) th payment, the loan is completely paid back.
2. Show that \(N ^ { 2 } - 81 N + 1620 = 0\)
3. Hence find the value of \(N\).

1. In an arithmetic series

• the first term is 16
• the 21 st term is 24
  1. Find the common difference of the series.
  2. Hence find the sum of the first 500 terms of the series.

1. The second, third and fourth terms of an arithmetic sequence are \(2 k , 5 k - 10\) and \(7 k - 14\) respectively, where \(k\) is a constant.

Show that the sum of the first \(n\) terms of the sequence is a square number.

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)

8 An arithmetic series has first term 9300 and 10th term 3900.

1. Show that the 20th term of the series is negative.
2. The sum of the first \(n\) terms is denoted by \(S\). Find the greatest value of \(S\) as \(n\) varies.

14

1. Use the laws of logarithms to show that \(\log _ { 10 } 200 - \log _ { 10 } 20\) is equal to 1 . The first three terms of a sequence are \(\log _ { 10 } 20 , \log _ { 10 } 200 , \log _ { 10 } 2000\).
2. Show that the sequence is arithmetic.
3. Find the exact value of the sum of the first 50 terms of this sequence.

7 A sequence is defined by the recurrence relation \(\mathrm { u } _ { \mathrm { k } + 1 } = \mathrm { u } _ { \mathrm { k } } + 5\) with \(\mathrm { u } _ { 1 } = - 2\).

1. Write down the values of \(u _ { 2 } , u _ { 3 }\), and \(u _ { 4 }\).
2. Explain whether this sequence is divergent or convergent.
3. Determine the value of \(u _ { 30 }\).
4. Determine the value of \(\sum _ { \mathrm { k } = 1 } ^ { 30 } \mathrm { u } _ { \mathrm { k } }\).

5 The first \(n\) terms of an arithmetic series are \(17 + 28 + 39 + \ldots + 281 + 292\).

1. Find the value of \(n\).
2. Find the sum of these \(n\) terms.

2. The sum of an arithmetic series is $$\sum _ { r = 1 } ^ { n } ( 80 - 3 r ) .$$

1. Write down the first two terms of the series.
2. Find the common difference of the series. Given that \(n = 50\),
3. find the sum of the series.