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

2. 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

3. 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

4. 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

5. 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 = -16
Explicit Formula
Recursive formula
Common difference = -11
A1 = 28
An = A1 + (n-1)d
An = 28 + (n-1)-11
An = n – 11
An = -11n +14
A1 = 3 An = A(n-1) + d A1 = 28 An = A(n-1) -11

6. 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 $ 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 = (n-1)-1.75
An = n
An = n
An = – 1.75(15) = – = $75.50 You will have $75.50 left on the pass after riding 15 times.
$100 pass:
0 = n
= -1.75n
58  n You can take about 58 bus rides for $100.

7. 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 = n-2
An = n
Common difference = -5
A1 = 21
An = 21 + (n-1)-5
An = 21 – 5n + 5
An = 26 – 5n

8. 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