Sequences and Series

1. Sequences and Series A sequence is an ordered list of numbers where each term is obtained according to a fixed rule. A series, or progression, is a sum. The terms of which form a sequence. The nth term of a sequence is often denoted Un, so that, for example, U is the first term. A sequence can be defined by a recurrence relation where Un+1 is given as a function of lower, earlier terms.

2. A first – order recurrence relation is where Un+1=rUn + d, where r and d are constants. This relation is linear. A sequence can be defined by a formula for Un, given as a function. Un = f(n) Being given the first few terms of a sequence is not enough to identify the sequence.

3. For example. Identify the next term in the sequence 1, 2, 3, …, …, Possible answers include: If however we also know that the sequence is generated by a first order linear recurrence relation, then we know

5. Example 1 Find the first order linear recurrence relation when: U3 = 7, U4 = 15 and U5 = 31.

6. Note Given a relation When this repetition happens, Un is referred to as a fixed point. In this case, for any other value of Un, the relation generates values that move away or diverge from the value of 2. Un =2 is an unstable fixed point.

7. Given the relation , then if for some value of n, Un = 4, the sequence would proceed 4, 4, 4, 4, …… If any other value of Un is used apart from 4, the relation generates terms whose value moves towards or converges on 4. Un = 4 is a stable fixed point, often referred to as the limit of the recurrence relation. In general, for the relation , we have a fixed point when

8. Arithmetic Sequences If a sequence is generated so that, for all n, then the sequence is known as an arithmetic sequence. The constant d is referred to as the common difference. This is a first order linear recurrence relation. Traditionally, U1 is represented by the letter a: U1 = a. This can be proved by induction – LATER !!!

9. Find the nth term • The 10th term of the arithmetic sequence 6, 11, 16, ……..

10. b) Find the arithmetic sequence for which U3 = 9 and U7 = 17. Subtracting gives: Substituting gives:

11. c) Given the arithmetic sequence 2, 8, 14, 20, …. For what value of n is Un = 62?

12. Page 117 Exercise 2A Questions 1 a, c, d 2 a to e 3, 4, 6. TJ Exercise 1 Questions 1 to 3

13. The Sum to n Terms on an Arithmetic Sequences Proof Adding

14. Find the sum of the first 15 terms of the arithmetic sequence which starts 3, 8, 13, 18, ……….

15. When does the sum of the arithmetic sequence which starts 2, 10, 18, 26,…. First exceed 300? We require Solving we get

16. The sum of the first four terms of an arithmetic sequence is 26. The sum of the first twelve terms is 222. What is the sum of the first 20 terms? NOTE

17. Page 120 Exercise 3A Questions 1, 3, 4, 5 and 8 TJ Exercise 1 Questions 4 to 9

18. Geometric Sequences If a sequence is generated so that for all then the sequence is known as a geometric sequence. The constant r is referred to as the common ratio. The nth term:

19. a) Find the nth term and the 10th term of the geometric sequence 3, 12, 48, ……. b) Find the geometric sequence whose 3rd term is 18 and whose 8th term is 4374 Substituting gives

20. c) Given the geometric sequence 5, 10, 20, 40,…… find the value of n for which

21. Page 123 Exercise 4A Questions 1 a - e, 2, 3, 5 and 7 TJ Exercise 2A Questions 1 to 4

22. The Sum to n Terms of a Geometric Sequence PROOF Multiplying by r: Subtracting:

23. a) Find the sum to 6 terms of the geometric sequence whose first term is 6 and whose common ratio is 1.5.

24. b) A geometric sequence starts 12, 15, 18.75,…… What is the smallest value of n for which Sn>100?

26. A geometric series is such that S3 = 14 and S6 = 126. • Identify the series. Dividing we get:

27. Page 127 Exercise 5a Questions 1, 2, 3, 4, TJ Exercise 2A Questions 5 to 7

28. Infinite Series, Partial Sums, Sum to infinity. An Infinite series is a series which has an infinite number of terms. When we have an infinite series then Sn is defined as the sum to n terms of that series. Such a sum is referred to as a partial sum of the series. If the partial sum, Sn, tends towards a limit as n tends to infinity, then the limit is called the sum to infinity of the series.

29. Arithmetic Series The sum to infinity for an arithmetic series is undefined.

30. Geometric Series

31. Find the sum to infinity of the geometric series • 24 + 12 + 6 + …….. If it exists.

32. b) Express the recurring decimal 0.121212…… as a vulgar fraction.

33. c) Given that 12 and 3 are two adjacent terms of an infinite geometric progression with find the first term. Hence the first term is 48.

34. Page 131 Exercise 6A Questions 1 to 4 and 7 T.J. Exercise 2B

35. Expanding (1-x)-1 and Related Functions Remember:

36. This is a geometric series with first term 1 and common ration r.

37. Now Consider This is a geometric series with common ratio

38. a) Expand in ascending powers of x giving the first four terms.

39. b) Expand giving the first four terms. c) Evaluate to 4 decimal places.

40. d) Expand in ascending powers of x giving the first four terms

41. Page 134 Exercise 7A Questions 2, 4. Page 134 Exercise 7B Questions 2, 5. TJ Exercise 3

42. The Sequence and Limit of

44. The Sigma Notation (i.e. the sum of all k2 for k = 1 to k = n) In general is the series with the first term f(1), second term f(2), third term f(3) and last term f(n) Summation of a Series The sigma notation is used as a more concise way of writing a series. e.g. 12 + 22 + 32 + 42 + 52 +…………+n2 can be written more concisely as

45. a) Write the following series in full. and so on to k = 10

46. b) Write the following series in full. and so on to k = 4

47. c) Express the following in  notation. 1+4+7+10+…….+298

48. Summation of a Series The sums of certain finite series can be found by a number of methods Proof: We can use this to help evaluate many summation series.