CHAPTER 6 Sequences and Series Around 1260 AD

CHAPTER 6 Sequences and Series

Around 1260 AD, the Kurdish historian Ibn Khallik¯an recorded the following story about Sissa ibn Dahir and a chess game against the Indian King Shihram. (The story is also told in the Legend of the Ambalappuzha Paal Payasam, where the Lord Krishna takes the place of Sissa ibn Dahir, and they play a game of chess with the prize of rice grains rather than wheat. ) King Shihram was a tyrant king, and his subject Sissa ibn Dahir wanted to teach him how important all of his people were. He invented the game of chess for the king, and the king was greatly impressed. He insisted on Sissa ibn Dahir naming his reward, and the wise man asked for one grain of wheat for the first square, two grains of wheat for the second square, four grains of wheat for the third square, and so on, doubling the wheat on each successive square on the board. The king laughed at first and agreed, for there was so little grain on the first few squares. By halfway he was surprised at the amount of grain being paid, and soon he realized his great error: that he owed more grain than there was in the world. a. How can we describe the number of grains of wheat for each square? b. What expression gives the number of grains of wheat for the nth square? • Find the total number of grains of wheat the king owed.

In mathematics it is important that we can: recognize a pattern in a set of numbers, describe the pattern in words, and continue the pattern

A number sequence is an ordered list of numbers defined by a rule. The numbers in the sequence are said to be its members or its terms. A sequence which continues forever is called an infinite sequence. A sequence which terminates is called a finite sequence.

4. Write down the first four terms of the sequence if you start with a. 45 and subtract 6 each time b. 96 and divide by 2 each time.

4. Write down the first four terms of the sequence if you start with a. 45 and subtract 6 each time 45, 39, 33, 27 b. 96 and divide by 2 each time. 96, 48, 24, 12

5. For each of the following write a description of the sequence and find the next 2 terms: a. 96, 89, 82, 75, . . b. 50 000, 10 000, 2000, 400, . .

5. For each of the following write a description of the sequence and find the next 2 terms: a. 96, 89, 82, 75, . . start with 96 and subtract 7 each time 68, 61 b. 50 000, 10 000, 2000, 400, . . start with 50, 000 and multiply 1/5 each time 80, 16

Sequences may be defined in one of the following ways: • listing all terms (of a finite sequence) • listing the first few terms and assuming that the pattern represented continues indefinitely • giving a description in words • using a formula which represents the general term or nth term.

The general term or nth term of a sequence is represented by a symbol with a subscript, for example un, Tn, tn, or An. The general term is defined for n = 1, 2, 3, 4, . . {un} represents the sequence that can be generated by using un as the nth term.

8. Consider the sequence defined by un = 2 n+ 5. Find the first four terms of the sequence

8. Consider the sequence defined by un = 2 n+ 5. Find the first four terms of the sequence. n = 1 n = 2 n = 3 n = 4 u 1 = 2(1) + 5 = 7 u 2 = 2(2) + 5 = 9 u 3 = 2(3) + 5 =11 u 4 = 2(4) + 5 = 13

9. Evaluate the first five terms of the sequence,

9. Evaluate the first five terms of the sequence,

10. Evaluate the first five terms of the sequence, .

10. Evaluate the first five terms of the sequence.

11. An arithmetic sequence is a sequence in which each term differs from the previous one by the same fixed number. It can also be referred to as an arithmetic progression.

12. {un} is arithmetic un+1 – un = d for all positive integers n where constant called the common difference.

13. For an arithmetic sequence with first term u 1 and common difference d the general term or nth term is un = u 1 + (n – 1)d.

14. Find the 10 th term of each of the following arithmetic sequences 101, 97, 93, 89, . .

14. Find the 10 th term of each of the following arithmetic sequences 101, 97, 93, 89, . . un = u 1 + (n – 1)d. u 10 = 101 + (10 – 1)(-4) = 65

15. Find the 15 th term of each of the following arithmetic sequences: a, a + d, a+ 2 d, a + 3 d, . .

15. Find the 15 th term of each of the following arithmetic sequences: a, a + d, a+ 2 d, a + 3 d, . . u 15 = a + (15 – 1)d = a + 14 d

16. Consider the sequence 87, 83, 79, 75, 71, . . a. Show that the sequence is arithmetic. b. Find the formula for its general term. c. Find the 40 th term. d. Which term of the sequence is -297?

16. Consider the sequence 87, 83, 79, 75, 71, . . a. Show that the sequence is arithmetic. 83 – 87 = 79 – 83 = 75 – 79 = 71 – 75 = -4 b. Find the formula for its general term. un = 87 + (n – 1)(-4) = 87 – 4 n + 4 = 91 – 4 n c. Find the 40 th term. un = 91 – 4(40) = -69 d. Which term of the sequence is -297? -297 = 91 – 4 n n = 97

17. A sequence is defined by. a. Prove that the sequence is arithmetic. b. Find u 1 and d. c. Find u 75. d. For what values of n are the terms of the sequence less than -200?

18. Find k given the consecutive arithmetic terms: a. k +1, 2 k + 1, 13 b. k – 1, 2 k + 3, 7 – k

19. Find the general term un for an arithmetic sequence with: a. u 7 = 41 and u 13 = 77 b. seventh term 1 and fifteenth term -39

20. A luxury car manufacturer sets up a factory for a new model. In the first month only 5 cars are produced. After this, 13 cars are assembled every month. a. List the total number of cars that have been made in the factory by the end of each of the first six months. b. Explain why the total number of cars made after n months forms an arithmetic sequence. c. How many cars are made in the first year? d. How long is it until the 250 th car is manufactured?

20. A luxury car manufacturer sets up a factory for a new model. In the first month only 5 cars are produced. After this, 13 cars are assembled every month. a. List the total number of cars that have been made in the factory by the end of each of the first six months. 5, 18, 31, 44, 57, 70 b. Explain why the total number of cars made after n months forms an arithmetic sequence. constant difference, d = 13 c. How many cars are made in the first year? u 12 = 5 + (12 – 1)(13) = 148 d. How long is it until the 250 th car is manufactured? 250 = 13 n – 8 n = 19. 84 20

21. A sequence is geometric if each term can be obtained from the previous one by multiplying by the same non-zero constant. A geometric sequence is also referred to as a geometric progression

{un } is geometric for all positive integers n constant called the common ratio.

For a geometric sequence with first term u 1 and common ratio r, the general term or nth term is un = u 1 r n – 1.

24. For the geometric sequence with first two terms given, find b and c: 2, 6, b, c, . .

24. For the geometric sequence with first two terms given, find b and c: 2, 6, b, c, . . common ratio = 6/2 = 3 b = 6 • 3 = 18 c = 18 • 3 = 54

25. Find the 9 th term in each of the following geometric sequences: 12, 18, 27, . .

26. a. Show that the sequence 12, -6, 3, -3/2 , . . is geometric.

26. a. Show that the sequence 12, -6, 3, -3/2 , . . is geometric.

b. Find un and hence write the 13 th term as a rational number.

26. b. Find un and hence write the 13 th term as a rational number.

27. Find k given that the following are consecutive terms of a geometric sequence k, k + 8, 9 k

28. Find the general term un of the geometric sequence which has: u 3 = 8 and u 6 = -1

A nest of ants initially contains 500 individuals. The population is increasing by 12% each week. a. How many ants will there be after: i 10 weeks ii 20 weeks? b. Use technology to find how many weeks it will take for the ant population to reach 2000.

We can calculate the value of a compounding investment using the formula un+1 = u 1 x r n where u 1 = initial investment r = growth multiplier for each period n = number of compounding periods un+1 = amount after n compounding periods.

31 a. What will an investment of $3000 at 10% p. a. compound interest amount to after 3 years? b. How much of this is interest?

32. How much compound interest is earned by investing $80 000 at 9% p. a. if the investment is over a 3 year period?

33. What initial investment is required to produce a maturing amount of $15 000 in 60 months’ time given a guaranteed fixed interest rate of 5. 5% p. a. compounded annually?

A series is the sum of the terms of a sequence. For the finite sequence {un} with n terms, the corresponding series is u 1 + u 2 + u 3 + …+ un. The sum of this series is Sn = u 1+u 2+u 3+…+un and this will always be a finite real number. For the infinite sequence {un} , the corresponding series is u 1 + u 2 + u 3 + … + un + … In many cases, the sum of an infinite series cannot be calculated. In some cases, however, it does converge to a finite number.

Sigma Notation u 1 + u 2 + u 3 + … + un

Properties of Sigma Notation

39. For the following sequences: i. write down an expression for Sn ii. find S 5. a. 3, 11, 19, 27, . . b. 12, 6, 3, 1. 5, . .

39. For the following sequences: i. write down an expression for Sn ii. find S 5. a. 3, 11, 19, 27, . .

40. Expand evaluate:

40. Expand evaluate:

41. For un = 3 n – 1, write u 1 + u 2 + u 3 +. . . + u 20 using sigma notation and evaluate the sum.

41. For un = 3 n – 1, write u 1 + u 2 + u 3 +. . . + u 20 using sigma notation and evaluate the sum.

42. An arithmetic series is the sum of the terms of an arithmetic sequence.

44. Find the sum of: 3 + 7 + 11+ 15 +. . . to 20 terms 50 + 48. 5 + 47 + 45. 5. . . to 80 terms

44. Find the sum of: 3 + 7 + 11+ 15 +. . . to 20 terms

50 + 48. 5 + 47 + 45. 5. . . to 80 terms

45. Find the sum of: 5 + 8 + 11+ 14 +. . . + 101

45. Find the sum of: 5 + 8 + 11+ 14 +. . . + 101

46. Evaluate the arithmetic series:

46. Evaluate the arithmetic series:

47. A soccer stadium has 25 sections of seating. Each section has 44 rows of seats, with 22 seats in the first row, 23 in the second row, 24 in the third row, and so on. How many seats are there in: a. row 44 of one section b. each section c. the whole stadium?

48. Three consecutive terms of an arithmetic sequence have a sum of 12 and a product of -80. Find the terms. Hint: Let the terms be x – d, x, and x + d.

49. A geometric series is the sum of the terms of a geometric sequence.

51. Find the sum of the following series: 12+ 6 + 3 + 1. 5 +. . . to 10 terms . . . 20 terms

52. Find a formula for Sn, the sum of the first n terms of the series: 0. 9 + 0. 009 + 0. 0009 +. . .

53. A geometric sequence has partial sums S 1 = 3 and S 2 = 4. a. State the first term u 1. b. Calculate the common ratio r. c. Calculate the fifth term u 5 of the series.

54. Evaluate these geometric series:

55.

56. Find the sum of each of the following infinite geometric series: a. 18 + 12 + 8 + 16/3 +. . . b. 18. 9 – 6. 3 + 2. 1 – 0. 7 +. . .

56. Find the sum of each of the following infinite geometric series: a. 18 + 12 + 8 + 16/3 +. . . b. 18. 9 – 6. 3 + 2. 1 – 0. 7 +. . .

57. Find the following