Sequences Ordered Patterns

8/31/2013 Sequences 2 The art of asking the right questions in mathematics is more important than the art of solving them − Georg Cantor ( )

8/31/2013 Sequences 3 What is a sequence? A pattern of objects arranged in an ordering corresponding to the ordering of the natural numbers Definitions: An infinite sequence is a function whose domain is the set of natural numbers A finite sequence is a function with domain D = { 1, 2, 3,..., n } for some positive integer n Sequences

8/31/2013 Sequences 4 ABCD ABCD ABCD Examples ABCDABCDABCD What are the next four characters in each of the above sequences? Identifying the pattern in the sequence allows for prediction of later values Sequences ABCD repeated group... odd natural numbers... prime numbers... multiples of 9... Fibonacci sequence

8/31/2013 Sequences 5 Terminology and Notation Applications of sequences generally do not require graphing, so the x-y notation is dropped Since the domain is the set of natural numbers, each domain element is an integer n Functional value is then written f(n) = a n Sequences As Functions

8/31/2013 Sequences 6 Terminology and Notation Range elements a n are called terms Terms are any kind of objects Examples: mile markers, fence posts, customers, integers, database records Terms can be arranged in “sequential” order via the subscript n – a sort of number tag Common notation: Sequences As Functions { a n }{ a n } n=1 k

8/31/2013 Sequences 7 { } (, ) 1 a1a1 Sequences As Functions As a Function Relates each member n of the domain with exactly one term a n in the range Domain = N Range 1 a1a1 2 a2a2 S = Questions: Is S a relation ? YES Is S a function ?YES 3 a3a3, … (, ) 3 a3a3,, 2 a2a2

8/31/2013 Sequences 8 Sequences As Functions Example Consider the following sequence Sequence is random – no recursion ! x        a 1 (-14) a 2 (0) a 3 (-37.5) a 4 (38) a 5 (5) a 6 (12) a 7 (-24) a 8 (26)  a 9 (44) { a n } n=1 9 = -14, 0, -37.5, 38, 5, 12, -24, 26, n

8/31/2013 Sequences 9 Recursive Sequences What is recursion ? Application of a repetitive pattern for generating successive terms Patterns generate a new term based on the value of its predecessors and a rule Additive – arithmetic sequences Multiplicative – geometric sequences Generalized Fibonacci & other patterns Not all sequences are recursive, e.g. random sequences with no general term

8/31/2013 Sequences 10 Sequences As Patterns Finding the General Term Suppose we start with the sequence a 1 = 2, a 2 = 4, a 3 = 8, a 4 = 16 What is the general term a n for any positive integer n ? Note that we can write the sequence as: a 1 = 2 1, a 2 = 2 2, a 3 = 2 3, a 4 = 2 4, …

8/31/2013 Sequences 11 Sequences As Patterns Finding the General Term Note that we can write the sequence as: a 1 = 2 1, a 2 = 2 2, a 3 = 2 3, a 4 = 2 4, … So we should have: a n = 2 n as the general term If f is the sequence function, then f(n) = a n = 2 n is the general term of the sequence

8/31/2013 Sequences 12 Example 1 Given: a 1 = 2, a 2 = 5, a 3 = 8, a 4 = 11 Find the n th term a n Note that a 2 – a 1 = 5 – 2 = 3 a 3 – a 2 = 8 – 5 = 3 a 4 – a 3 = 11 – 8 = 3 Thus that is, the common difference is 3 Finding the General Term a n+1 – a n = 3 for n ≥ 1

8/31/2013 Sequences 13 Example 1 a n+1 – a n = 3 for n ≥ 1 a n+1 = a n + 3 So Finding the General Term a 2 = a = a 1 + 1(3) = 5 a 3 = a = 8 = a 1 + 2(3) a 4 = a = a 1 + 3(3) = 11 a 5 = a = a 1 + 4(3) = 14 = a 1 + (n – 1)3 = 2 + (n – 1)3 anan

8/31/2013 Sequences 14 Example 1 Finding the General Term = 2 + (n – 1)3 anan Question: Does this work for a 1, a 2, a 3 ? Question: What is a 21 ? Note: The general form of a n was found inductively from specific values = 2 + (21 – 1)3 a 21 = 62 Note: We find the specific value of a 21 deductively from the general form

8/31/2013 Sequences 15 Finding Sequence Terms From the General to the Specific Example 2 Find the first four terms for: a 2 = 3(2 – 1) + 5 = 8 a 4 = 3(4 – 1) + 5 = 14 a n = 3(n – 1) + 5 a 1 = 3(1 – 1) + 5 = 5 a 3 = 3(3 – 1) + 5 = 11 What is a 21 ? Question:

8/31/2013 Sequences 16 Finding Sequence Terms From the General to the Specific Example 3 Find the first four terms for: n 1 (–1) n = –1 2 1 = Question: What is a 21 ? a1a1 = 1 1 (–1) 1 a2a2 = 2 1 (–1) 2 a3a3 = 3 1 (–1) 3 = 3 1 – 4 1 = a4a4 = 4 1 (–1) 4

8/31/2013 Sequences 17 Recursive Sequences Finding Terms from Preceding Terms Consider the sequence Here we have a 1 = 1, a 2 = 1, a 3 = 2, a 4 = 3, a 5 = 5,... So, how are these related? Well... note that, a 4 = a 3 + a 2, a 5 = a 4 + a 3,... a 3 = a 2 + a 1

8/31/2013 Sequences 18 Recursive Sequences Finding Terms from Preceding Terms, a 4 = a 3 + a 2, a 5 = a 4 + a 3,... a 3 = a 2 + a 1 Generally appears that, starting with n = 3, a n = a n–1 + a n–2 Functionally, it appears that f(n) = f(n – 1) + f(n – 2) for n ≥ 3 This is a recursive function... in this case the basic Fibonacci Sequence

8/31/2013 Sequences 19 Recursive Sequences Examples 1. a 1 = –1 and a n = a n–1 + 4 Find the first four terms a 1 = –1, a 2 = a = 3, a 3 = a = 7, a 4 = a = 11 Notice anything about the graph ? n f(n) = a n ● ● ● ●

8/31/2013 Sequences 20 Recursive Sequences Examples 1. a 1 = –1 and a n = a n–1 + 4 n f(n) = a n ● ● ● ● What is ∆f from n to n + 1 ? ∆f ∆n = a n+1 – a n (n + 1) – n = = If we were to allow n = 0, what would f(0) be ?

8/31/2013 Sequences 21 Recursive Sequences Examples 2. a 1 = 0, a 2 = 1 and a n = 2 a n–1 + a n–2 Find the first five terms a 3 = 2 a 2 + a 1 = 2  Is f(n) = a n a linear function ? n f(n) = a n ● ● ● ● ● a 1 = 0, a 2 = 1,, … a 4 = 2 a 3 + a 2 = 5, a 5 = 2 a 4 + a 3 = 12

8/31/2013 Sequences 22 Recursive Sequences Examples 3. a n = 3 a n–1 and Find the first five terms n f(n) = a n a1a = 3 =, a 2 = 3a13a1 = 9 1 a 3 = 3 a 2 a 4 = 3 a 3 = 1 a 5 = 3 a 4 = 3 a n = 3 n–1 a 1 Is f(n) = a n Question: = 3(3 a 1 ) = 3(3 2 a 1 ) = 3 3 a 1 = 3 4 a 1 ● ● ● ● ● a1a = = 3 1 = 3 2 a 1 a linear function ?

8/31/2013 Sequences 23 Arithmetic Sequences Definition An arithmetic sequence is a function defined on the set of positive integers of form f(n) = a n = a n–1 + d where d is the common difference

8/31/2013 Sequences 24 Arithmetic Sequences Arithmetic Sequence: a n = a n–1 + d Clearly F F B For n = 1, a 1 is given independently a n – a n–1 = d for n ≥ 2 By induction, a n = a 1 + (n – 1)d for all n > 1 For n > 1, a n is computed recursively for each successive n

8/31/2013 Sequences 25 Arithmetic Sequences Arithmetic Sequence: a n = a n–1 + d Example: f(1) = a 1 = – 4, d = 3 a 2 = a = – = –1 a 3 = a = 2 = ( a 1 + 3) + 3 = 5 a 4 = a = ( ( a 1 + 2(3) ) + 3 = a 1 + 2(3) = a 1 + 3(3) a n = a 1 + (n – 1)3 = – 4 + 3(n – 1) a 5 = a = ( ( a 1 + 3(3) ) + 3 = a 1 + 4(3) = 8 Question: Is f(n) linear ?

8/31/2013 Sequences 26 Arithmetic Sequences as Functions Given arithmetic sequence { a n } where a n = a n–1 + d = a 1 + (n – 1)d Function f(n) is f(n) = a n = a 1 + (n – 1)d f(n) can be written as f(k) = a 1 + kd = dk + a 1 where k = n – 1 Arithmetic Sequences

8/31/2013 Sequences 27 Arithmetic Sequences as Functions f(k) = a 1 + kd = dk + a 1 where k = n – 1 Arithmetic Sequences The rate of change of f(k) is d and f(0) = a 1 Thus f(k) = dk + a 1 a linear function with slope d and vertical intercept (0, a 1 )

8/31/2013 Sequences 28 k f(k)           Example Arithmetic Sequences n …. k …. f(k) …. f(k) = 3k + 5 n k f(k) Given arithmetic sequence { a n } with a 1 = 5 and d = 3, map the sequence function f(n) = dk + a 1

8/31/2013 Sequences 29 Geometric Sequences Definition A geometric sequence is a function defined on the set of positive integers of form f(n) = a n = r a n–1 where r is the common ratio and a 1 = c is a constant

8/31/2013 Sequences 30 Geometric Sequences A geometric sequence is a function defined on the set of positive integers of form f(n) = a n = r a n–1 By induction we can show that r a n–1 = r n–1 a 1 anan a n–1 = r, the ratio of successive terms Note:

8/31/2013 Sequences 31 Geometric Sequences Example, r = 3 = a 2a 2 a1a = = 1 = anan a1a1 3 n–1 Is f(n) = a n a linear function ? Question: f(1) = a = = = = a3a3 a2a2 3 = a1a1 3 3(3( ) = 3 2 a13 2 a1 = 3 n–1 1 9 By induction

8/31/2013 Sequences 32 Geometric Sequences Geometric Sequences as Functions Given geometric sequence { a n } where a n = r a n–1 = r n–1 a 1 The function f(n) = a n = r n–1 a 1 can be written, with k = n – 1, as f(k) = r k a 1 = a 1 r k making f(k) an exponential function The rate of change of f(k) is r k (r – 1) a 1 and so is never constant

8/31/2013 Sequences 33 k f(k) Geometric Sequences Example … … ….         f(k) = 3(2 k ) Given geometric sequence { a n } with a 1 = 3 and r = 2 Sequence function: f(n) = a 1 (r n–1 ) or f(k) = a 1 (r k ) for k = n – 1 n k f(k)

8/31/2013 Sequences 34 Retrospective Sequences in Review Arithmetic Sequence Successive terms with common difference Sequence function is linear Geometric Sequence Successive terms with common ratio Sequence function is exponential Other Sequences Many recursion patterns possible Random sequences without pattern

8/31/2013 Sequences 35 Think about it !

8/31/2013 Sequences 36 Examples ABCDABCDABCD Identifying the pattern in the sequence allows for prediction of later values What are the next four characters in each of the above sequences? ABCD ABCD ABCD Sequences ABCD repeated group... odd natural numbers... prime numbers... multiples of 9... Fibonacci sequence