1.04h Arithmetic sequences: nth term and sum formulae

5. A company that owned a silver mine

• extracted 480 tonnes of silver from the mine in year 1
• extracted 465 tonnes of silver from the mine in year 2
• extracted 450 tonnes of silver from the mine in year 3
  and so on, forming an arithmetic sequence.
  1. Find the mass of silver extracted in year 14

After a total of 7770 tonnes of silver was extracted, the company stopped mining. Given that this occurred at the end of year \(N\),

show that $$N ^ { 2 } - 65 N + 1036 = 0$$

Hence, state the value of \(N\).

1. In a large theatre there are \(n\) rows of seats, where \(n\) is a constant.

The number of seats in the first row is \(a\), where \(a\) is a constant.
In each subsequent row there are 4 more seats than in the previous row so that

• in the 2 nd row there are \(( a + 4 )\) seats
• in the 3rd row there are ( \(a + 8\) ) seats
• the number of seats in each row form an arithmetic sequence

Given that the total number of seats in the first 10 rows is 360

1. find the value of \(a\). Given also that the total number of seats in the \(n\) rows is 2146
2. show that $$n ^ { 2 } + 8 n - 1073 = 0$$
3. Hence
   1. state the number of rows of seats in the theatre,
   2. find the maximum number of seats in any one row.

A girl saves money over a period of 200 weeks. She saves 5 p in Week 1,7 p in Week 2, 9p in Week 3, and so on until Week 200. Her weekly savings form an arithmetic sequence.

1. Find the amount she saves in Week 200.
2. Calculate her total savings over the complete 200 week period.

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 $$\frac { 1 } { 2 } n [ 2 a + ( n - 1 ) d ]$$ 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 \(N\).
2. Find the value of \(N\) The company then plans to continue to make 600 mobile phones each week.
3. Find the total number of mobile phones that will be made in the first 52 weeks starting from and including week 1.
   
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1 The 20th term of an arithmetic progression is 10 and the 50th term is 70 .

1. Find the first term and the common difference.
2. Show that the sum of the first 29 terms is zero.

6 A sequence of terms \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { n } = 2 n + 5 , \quad \text { for } n \geqslant 1 .$$

1. Write down the values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\).
2. State what type of sequence it is.
3. Given that \(\sum _ { n = 1 } ^ { N } u _ { n } = 2200\), find the value of \(N\).

1 A sequence \(S\) has terms \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) defined by $$u _ { n } = 3 n - 1 ,$$ for \(n \geqslant 1\).

1. Write down the values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\), and state what type of sequence \(S\) is.
2. Evaluate \(\sum _ { n = 1 } ^ { 100 } u _ { n }\).

6

1. John aims to pay a certain amount of money each month into a pension fund. He plans to pay \(\pounds 100\) in the first month, and then to increase the amount paid by \(\pounds 5\) each month, i.e. paying \(\pounds 105\) in the second month, \(\pounds 110\) in the third month, etc. If John continues making payments according to this plan for 240 months, calculate
   1. how much he will pay in the final month,
   2. how much he will pay altogether over the whole period.
   3. Rachel also plans to pay money monthly into a pension fund over a period of 240 months, starting with \(\pounds 100\) in the first month. Her monthly payments will form a geometric progression, and she will pay \(\pounds 1500\) in the final month. Calculate how much Rachel will pay altogether over the whole period.

7

1. In an arithmetic progression, the first term is 12 and the sum of the first 70 terms is 12915 . Find the common difference.
2. In a geometric progression, the second term is - 4 and the sum to infinity is 9 . Find the common ratio.

6 A sequence is given by $$\begin{gathered} a _ { 1 } = 4 \\ a _ { r + 1 } = a _ { r } + 3 \end{gathered}$$ Write down the first 4 terms of this sequence.
Find the sum of the first 100 terms of the sequence.

8 The 7th term of an arithmetic progression is 6. The sum of the first 10 terms of the progression is 30. Find the 5th term of the progression.

2 The \(n\)th term of an arithmetic progression is \(6 + 5 n\). Find the sum of the first 20 terms.

11 There is a flowerhead at the end of each stem of an oleander plant. The next year, each flowerhead is replaced by three stems and flowerheads, as shown in Fig. 11. \begin{figure}[h] \end{figure}

1. How many flowerheads are there in year 5?
2. How many flowerheads are there in year \(n\) ?
3. As shown in Fig. 11, the total number of stems in year 2 is 4, (that is, 1 old one and 3 new ones). Similarly, the total number of stems in year 3 is 13 , (that is, \(1 + 3 + 9\) ). Show that the total number of stems in year \(n\) is given by \(\frac { 3 ^ { n } - 1 } { 2 }\).
4. Kitty's oleander has a total of 364 stems. Find
   (A) its age,
   (B) how many flowerheads it has.
5. Abdul's oleander has over 900 flowerheads. Show that its age, \(y\) years, satisfies the inequality \(y > \frac { \log _ { 10 } 900 } { \log _ { 10 } 3 } + 1\).
   Find the smallest integer value of \(y\) for which this is true.

11

1. André is playing a game where he makes piles of counters. He puts 3 counters in the first pile. Each successive pile he makes has 2 more counters in it than the previous one.
   1. How many counters are there in his sixth pile?
   2. André makes ten piles of counters. How many counters has he used altogether?
2. In another game, played with an ordinary fair die and counters, Betty needs to throw a six to start. The probability \(\mathrm { P } _ { n }\) of Betty starting on her \(n\)th throw is given by $$P _ { n } = \frac { 1 } { 6 } \times \left( \frac { 5 } { 6 } \right) ^ { n - 1 }$$
   1. Calculate \(\mathrm { P } _ { 4 }\). Give your answer as a fraction.
   2. The values \(\mathrm { P } _ { 1 } , \mathrm { P } _ { 2 } , \mathrm { P } _ { 3 } , \ldots\) form an infinite geometric progression. State the first term and the common ratio of this progression. Hence show that \(\mathrm { P } _ { 1 } + \mathrm { P } _ { 2 } + \mathrm { P } _ { 3 } + \ldots = 1\).
   3. Given that \(\mathrm { P } _ { n } < 0.001\), show that \(n\) satisfies the inequality $$n > \frac { \log _ { 10 } 0.006 } { \log _ { 10 } \left( \frac { 5 } { 6 } \right) } + 1$$ Hence find the least value of \(n\) for which \(\mathrm { P } _ { n } < 0.001\).

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 \(\pounds 2621435\).

3 On his \(1 ^ { \text {st } }\) birthday, John was given \(\pounds 5\) by his Uncle Fred. On each succeeding birthday, Uncle Fred gave a sum of money that was \(\pounds 3\) more than the amount he gave on the last birthday.

1. How much did Uncle Fred give John on his \(8 { } ^ { \text {th } }\) birthday?
2. On what birthday did the gift from Uncle Fred result in the total sum given on all birthdays exceeding £200?

4 The first 3 terms of an arithmetical progression are 7, 5.9 and 4.8.
Find

1. the common difference,
2. the smallest value of \(n\) for which the sum to \(n\) terms is negative.

11 When Fred joined a computer firm his salary was \(\pounds 28000\) per annum. In each subsequent year he received an annual increase of \(12 \%\) of his previous year's salary.

1. State Fred's salary for each of his first 3 years with the company. State also the common ratio of the geometric sequence formed by his salaries.
2. How much did Fred earn in the tenth year?
3. Show that the total amount Fred earned over the ten years was between \(\pounds 400000\) and £500000.
4. When Fred joined the computer firm, his brother Archie set up a plumbing business. He earned \(\pounds 35000\) in his first year and each year earned \(\pounds d\) more than in the previous year. At the end of ten years, he had earned exactly the same total amount as Fred. Calculate the value of \(d\).

9. The first two terms of a geometric progression are 2 and \(x\) respectively, where \(x \neq 2\).

1. Find an expression for the third term in terms of \(x\). The first and third terms of arithmetic progression are 2 and \(x\) respectively.
2. Find an expression for the 11th term in terms of \(x\). Given that the third term of the geometric progression and the 11th term of the arithmetic progression have the same value,
3. find the value of \(x\),
4. find the sum of the first 50 terms of the arithmetic progression.

1. (i) Evaluate

$$\sum _ { r = 1 } ^ { 50 } ( 80 - 3 r )$$ (ii) Show that $$\sum _ { r = 1 } ^ { n } \frac { r + 3 } { 2 } = k n ( n + 7 )$$ where \(k\) is a rational constant to be found.

8. A store begins to stock a new range of DVD players and achieves sales of \(\pounds 1500\) of these products during the first month. In a model it is assumed that sales will decrease by \(\pounds x\) in each subsequent month, forming an arithmetic sequence. Given that sales total \(\pounds 8100\) during the first six months, use the model to

1. find the value of \(x\),
2. find the expected value of sales in the eighth month,
3. show that the expected total of sales in pounds during the first \(n\) months is given by \(k n ( 51 - n )\), where \(k\) is an integer to be found.
4. Explain why this model cannot be valid over a long period of time.

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 \(\pounds 2621435\).

3 The 11th term of an arithmetic progression is 1. The sum of the first 10 terms is 120. Find the 4th term.

4

1. André is playing a game where he makes piles of counters. He puts 3 counters in the first pile. Each successive pile he makes has 2 more counters in it than the previous one.
   1. How many counters are there in his sixth pile?
   2. André makes ten piles of counters. How many counters has he used altogether?
2. In another game, played with an ordinary fair die and counters, Betty needs to throw a six to start. The probability \(\mathrm { P } _ { n }\) of Betty starting on her \(n\)th throw is given by $$P _ { n } = \frac { 1 } { 6 } \times \left( \frac { 5 } { 6 } \right) ^ { n - 1 }$$
   1. Calculate \(\mathrm { P } _ { 4 }\). Give your answer as a fraction.
   2. The values \(\mathrm { P } _ { 1 } , \mathrm { P } _ { 2 } , \mathrm { P } _ { 3 } , \ldots\) form an infinite geometric progression. State the first term and the common ratio of this progression. Hence show that \(\mathrm { P } _ { 1 } + \mathrm { P } _ { 2 } + \mathrm { P } _ { 3 } + \ldots = 1\).
   3. Given that \(\mathrm { P } _ { n } < 0.001\), show that \(n\) satisfies the inequality $$n > \frac { \log _ { 10 } 0.006 } { \log _ { 10 } \left( \frac { 5 } { 6 } \right) } + 1$$ Hence find the least value of \(n\) for which \(\mathrm { P } _ { n } < 0.001\).