SEQUENCES AND SERIES

Transcription

1 A sequence is an ordered list of numbers. SEQUENCES AND SERIES Note, in this context, ordered does not mean that the numbers in the list are increasing or decreasing. Instead it means that there is a first member in the sequence, a second, a third and so on. The following are examples of sequences: 1, 2, 3, 4, 5, 1, 0,1, 0,1, 0,1, 1, 2, 3, 4, 5, 6, 1, 1 2, 1 3, 1 4, 1 5, 3,1, 4,1, 5, 9, 2, 6, A sequence can be graphed. The sequence defined by t n = n is graphed below: The sequence defined by t n = ( 1) n+1 is graphed below: The School For Excellence 2016 Summer School Unit 1 Specialist Mathematics Book 1 Page 13

2 A sequence can be defined in various ways. Usually we will define sequences by: 1. Forumula or function ( t n = f (n)) 2. Recurrence relation, where each term is defined in terms of the previous ones (for example, the Fibonacci sequence is often presented in this form). In the following exercises we will see examples of both. QUESTION 9 Write down the first four terms of the sequences defined by: (a) t n = 2n +1 (b) t n = (n 1) (c) t n = 1 2n +1 The School For Excellence 2016 Summer School Unit 1 Specialist Mathematics Book 1 Page 14

3 (d) t n+1 = 2t n +1, t 1 = 3 (e) t n+2 = t n+1 + t n, t 1 = t 2 =1 (f) t n+1 = nt n, t 1 = 1 The School For Excellence 2016 Summer School Unit 1 Specialist Mathematics Book 1 Page 15

4 ARITHMETIC SEQUENCES An arithmetic sequence is a sequence t 1,t 2,t 3, which can be written in recursive form as t n+1 = t n + d, t 1 = a. This gives us another way to write an arithmetic sequence: a, a + d, a + 2d, a + 3d, and so t n = a + (n 1)d. The value of d is the common difference. EXAMPLE 1 Determine the rule for the arithmetic sequence 3, 5, 7, 9, The first term is 3 and the common difference is 2. The rule for the sequence is t n = 3+ 2(n 1) EXAMPLE 2 What is 15 th term of the arithmetic sequence which has a first term of 5 and a common difference of 3? We have that: t 15 = 5+ (15 1) 3 = = 47 EXAMPLE 3 Determine the 20 th term of the arithmetic sequence which has t 2 = 10 and t 10 = 66. We can construct simultaneous equations: a + d =10 a + 9d = 66 Solving these equations gives d = 7 and a = 3. Therefore: t 20 = =136 The School For Excellence 2016 Summer School Unit 1 Specialist Mathematics Book 1 Page 16

5 QUESTION 10 Determine the 25 th term of the arithmetic sequence whose third term is 10 and whose tenth term is 59. The School For Excellence 2016 Summer School Unit 1 Specialist Mathematics Book 1 Page 17

6 ARITHMETIC SERIES A series is the sum of the terms of a sequence. For the arithmetic sequence t n = a + (n 1)d, n =1, 2, we will write the series which is the sum of the first n terms as: S n = a + a + d + a + 2d + + a + (n 1)d It is a fairly simple exercise to show that: S n = n ( 2a + (n 1)d) 2 QUESTION 11 Determine the sum of the first 20 terms of the arithmetic sequence which begins 5, 1,. The School For Excellence 2016 Summer School Unit 1 Specialist Mathematics Book 1 Page 18

7 QUESTION 12 Determine the number of terms n required to produce a sum of 60 in an arithmetic sequence whose first term is 2 and has a common difference of 5. The School For Excellence 2016 Summer School Unit 1 Specialist Mathematics Book 1 Page 19

8 GEOMETRIC SEQUENCES A sequence is called a geometric sequence if it is of the form: a, ar, ar 2, ar 3, The first term is a and the value of r is called the common ratio. The n th term is t n = ar n 1. EXAMPLE 1 The sequence defined by t n = 2 n 1, n =1, 2, 3, is a geometric sequence. The first few terms are 1, 2, 4,8,16,32, 64, EXAMPLE 2 The sequence whose n th term is 1 2 n 1, n =1, 2, 3, is a geometric sequence. The first terms of this sequence are 1, 1 2, 1 4, 1 8, The School For Excellence 2016 Summer School Unit 1 Specialist Mathematics Book 1 Page 20

9 QUESTION 13 The numbers n 2, n and n + 3 are consecutive terms of a geometric sequence. Find n and hence write down the next term. The School For Excellence 2016 Summer School Unit 1 Specialist Mathematics Book 1 Page 21

10 GEOMETRIC SERIES We can determine the sum of the first n terms of a geometric sequence as follows: Let S n = a + ar + ar 2 + ar n 1 Then rs n = ar + ar ar n 1 + ar n = ( a + ar + ar ar n 1 )+ ar n a = S n + ar n a Subtracting S n from both sides and factorising gives S n ( r 1)= ar n a ( ) S n = arn 1 r 1 1 rn = a 1 r Note that r 1. If r =1 then we have a trivial geometric series and the sum S n = na. QUESTION 14 (a) Determine the sum of the first 10 terms of the geometric sequence which begins 3, 1 3, The School For Excellence 2016 Summer School Unit 1 Specialist Mathematics Book 1 Page 22

11 (b) Determine the values of a and r if S 5 = 217 and S 7 = 889 and a,r > 0. The School For Excellence 2016 Summer School Unit 1 Specialist Mathematics Book 1 Page 23

12 (c) The 9 th term of a geometric sequence is 32 times the size of the 4 th term. If the second term is 6, find the sum of the first 7 terms of the sequence. The School For Excellence 2016 Summer School Unit 1 Specialist Mathematics Book 1 Page 24