Transcript Patterns and Sequences

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. 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
• Common difference = +7
• A1 = 3
An = A1 + (n-1)d
An = 3 + (n-1)7
An = 3 + 7n – 7
An = 7n -4
Recursive formula
A1 = first term
An = A(n-1) + d
A1 = 3
An = A(n-1) + 7
Arithmetic
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
• Common factor = -2
• A1 = 2
An = A1 (rn)
An =
2(-2n)
Recursive formula
A1 = first term
An = A(n-1) * r
A1 = 2
An = A(n-1)(-2)
Geometric
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
• Common difference = -11
• A1 = 28
An = A1 + (n-1)d
An = 28 + (n-1)-11
An = 28 - 11n – 11
An = -11n +14
Recursive formula
A1 = 3
An = A(n-1) + d
A1 = 28
An = A(n-1) -11
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 = 101.75 – 1.75(15)
• An = 100 - 1.75n + 1.75
An = 101.75 – 1.75(15)
• An = 101.75 - 1.75n
= 101.75 – 26.25
= $75.50 You will have
$75.50 left on the pass
$100 pass:
after riding 15 times.
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
• Common difference = +2
• A1 = 32
An = A(n-1) -5 ; A1 = 21
• Common difference = -5
• A1 = 21
An = 32 + (n-1)2
An = 32 + 2n-2
An = 30 + 2n
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
• Common difference = 10
• An = 76
An = 76
An = A(n-1) + 10
An = 1 + (n-1)-3
• Common difference = -3
• An = 1
An = 1
An = A(n-1) – 3