1.04i Geometric sequences: nth term and finite series sum

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.

5 The first three terms of a geometric progression are 4, 2, 1.
Find the twentieth term, expressing your answer as a power of 2.
Find also the sum to infinity of this progression.

2 A geometric progression has 6 as its first term. Its sum to infinity is 5 .
Calculate its common ratio.

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.

8 The second term of a geometric progression is 18 and the fourth term is 2 . The common ratio is positive. Find the sum to infinity of this progression.

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 \(\pounds 2621435\).

7 The first two terms of a geometric series are 5 and 4.
Find

1. the sum of the first 10 terms,
2. the sum to infinity.

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.

1 The common ratio of a geometric progression is - 0.5 . The sum of its first three terms is 15 . Find the first term.
Find also the sum to infinity.

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 \(d\).

7. A student completes a mathematics course and begins to work through past exam papers. He completes the first paper in 2 hours and the second in 1 hour 54 minutes. Assuming that the times he takes to complete successive papers form a geometric sequence,

1. find, to the nearest minute, how long he will take to complete the fifth paper,
2. show that the total time he takes to complete the first eight papers is approximately 13 hours 28 minutes,
3. find the least number of papers he must work through if he is to complete a paper in less than one hour.

1. Evaluate

$$\sum _ { r = 1 } ^ { 12 } \left( 5 \times 2 ^ { r } \right)$$

7. \includegraphics[max width=\textwidth, alt={}, center]{e4afa57d-5be3-42a6-ab35-39b0fdcc1681-2_364_666_1338_568} The diagram shows part of a design being produced by a computer program.
The program draws a series of circles with each one touching the previous one and such that their centres lie on a horizontal straight line. The radii of the circles form a geometric sequence with first term 1 mm and second term 1.5 mm . The width of the design is \(w\) as shown.

1. Find the radius of the fourth circle to be drawn.
2. Show that when eight circles have been drawn, \(w = 98.5 \mathrm {~mm}\) to 3 significant figures.
3. Find the total area of the design in square centimetres when ten circles have been drawn.

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.

7. The second and third terms of a geometric series are \(\log _ { 3 } 4\) and \(\log _ { 3 } 16\) respectively.

1. Find the common ratio of the series.
2. Show that the first term of the series is \(\log _ { 3 } 2\).
3. Find, to 3 significant figures, the sum of the first six terms of the series.

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 \(\pounds 2621435\).

2 The first term of a geometric series is 5.4 and the common ratio is 0.1.

1. Find the fourth term of the series.
2. Find the sum to infinity of the series.

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 \(\mathrm { P } _ { n } < 0.001\).

2 The first three terms of a geometric progression are 4, 2, 1.
Find the twentieth term, expressing your answer as a power of 2 .
Find also the sum to infinity of this progression.

4 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.

5 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 The terms of a sequence are given by $$\begin{aligned} u _ { 1 } & = 192 , \\ u _ { n + 1 } & = - \frac { 1 } { 2 } u _ { n } . \end{aligned}$$

1. Find the third term of this sequence and state what type of sequence it is.
2. Show that the series \(u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots\) converges and find its sum to infinity.