Patterns and Sequences To identify and extend patterns in sequences To represent arithmetic sequences using function notation

Vocabulary Sequence –an ordered list of numbers that often form a pattern Term of a sequence – each number in the sequence Arithmetic sequence – when the difference between each consecutive term is constant Common difference – the difference between the terms Geometric sequence – when consecutive terms have a common factor Recursive formula – a formula that related each term of the sequence to the term before it Explicit formula – a function rule that relates each term of the sequence to the term number

EX1: Describe a pattern in each sequence EX1: Describe a pattern in each sequence. What are the next two terms of each sequence? 3, 10, 17, 24, 31, … The patterns is to add 7 to the previous term. The next two terms are 31+7 = 38 and 38+7 = 45 Explicit Formula Recursive formula Common difference = +7 A1 = 3 An = A1 + (n-1)d An = 3 + (n-1)7 An = 3 + 7n – 7 An = 7n -4 A1 = first term An = A(n-1) + d A1 = 3 An = A(n-1) + 7 Arithmetic sequence

EX2: Describe a pattern in each sequence EX2: Describe a pattern in each sequence. What are the next two terms of each sequence? 2, -4, 8, -16, … The patterns is to multiply by -2 to the previous term. The next two terms are -16*-2 = 32 and 32*-2 = -64 Explicit Formula Recursive formula Common factor = -2 A1 = 2 An = A1 (rn) An = 2(-2n) A1 = first term An = A(n-1) * r A1 = 2 An = A(n-1)(-2) Geometric sequence

Try: Describe a pattern in each sequence Try: Describe a pattern in each sequence. What are the next two terms of each sequence? 28, 17, 6, … The patterns is to subtract 11 to the previous term. The next two terms are 6-11= -5 and -5-11 = -16 Explicit Formula Recursive formula Common difference = -11 A1 = 28 An = A1 + (n-1)d An = 28 + (n-1)-11 An = 28 - 11n – 11 An = -11n +14 A1 = 3 An = A(n-1) + d A1 = 28 An = A(n-1) -11

Ex3: A subway pass has a starting value of $100 Ex3: A subway pass has a starting value of $100. After one ride, the value of the pass is $98.25, after two rides, its value is $94.75. Write an explicit formula to represent the remaining value of the pass after 15 rides? How many rides can be taken with the $100 pass? Explicit formula 15 Rides: An = 100 + (n-1)-1.75 An = 100 - 1.75n + 1.75 An = 101.75 - 1.75n An = 101.75 – 1.75(15) = 101.75 – 26.25 = $75.50 You will have $75.50 left on the pass after riding 15 times. $100 pass: 0 = 101.75-1.75n -101.75 -101.75 -101.75 = -1.75n -1.75 -1.75 58  n You can take about 58 bus rides for $100.

Ex4: Writing an explicit function from a recursive function. An = A(n-1) + 2; A1 = 32 An = A(n-1) -5 ; A1 = 21 Common difference = +2 A1 = 32 An = 32 + (n-1)2 An = 32 + 2n-2 An = 30 + 2n Common difference = -5 A1 = 21 An = 21 + (n-1)-5 An = 21 – 5n + 5 An = 26 – 5n

Ex 5: Write a recursive formula from an explicit formula An = 76 + (n-1)10 An = 1 + (n-1)-3 Common difference = 10 An = 76 An = A(n-1) + 10 Common difference = -3 An = 1 An = A(n-1) – 3