Transcript Section 9.6 Sequences

Section 9.6 Sequences
Def: A sequence is a list of items occurring in a
specified order. Items may be numbers, letters,
objects, movements, etc.
Def: A sequence is a list of items occurring in a
specified order. Items may be numbers, letters,
objects, movements, etc.
Examples:
• clap clap stomp clap clap stomp clap clap stomp
Def: A sequence is a list of items occurring in a
specified order. Items may be numbers, letters,
objects, movements, etc.
Examples:
• clap clap stomp clap clap stomp clap clap stomp
A
A
B
A
A
B
A
A
B
Def: A sequence is a list of items occurring in a
specified order. Items may be numbers, letters,
objects, movements, etc.
Examples:
• clap clap stomp clap clap stomp clap clap stomp
A
A
B
A
A
B
A
A
B
• 1, 2, 3, 4, 5, 6, 7, ….
Def: A sequence is a list of items occurring in a
specified order. Items may be numbers, letters,
objects, movements, etc.
Examples:
• clap clap stomp clap clap stomp clap clap stomp
A
A
B
A
A
B
A
A
B
• 1, 2, 3, 4, 5, 6, 7, ….
•
Def: A sequence is a list of items occurring in a
specified order. Items may be numbers, letters,
objects, movements, etc.
Examples:
• clap clap stomp clap clap stomp clap clap stomp
A
A
B
A
A
B
A
A
B
• 1, 2, 3, 4, 5, 6, 7, ….
•
•
……
3
6
9
Repeating Patterns
Each sequence is made up of a unit that is repeated a certain number
of times or infinitely. We consider the unit to be the smallest repeated
portion.
Repeating Patterns
Each sequence is made up of a unit that is repeated a certain number
of times or infinitely. We consider the unit to be the smallest repeated
portion.
Repeating Patterns
Each sequence is made up of a unit that is repeated a certain number
of times or infinitely. We consider the unit to be the smallest repeated
portion.
……
Repeating Patterns
Each sequence is made up of a unit that is repeated a certain number
of times or infinitely. We consider the unit to be the smallest repeated
portion.
……
Repeated Patterns Example Problem
Ex 1: What is the 75th object in the following sequence?
……
Growing Patterns:
Def: An arithmetic sequence begins with any number as the
1st entry, but each subsequent entry is obtained by adding or
subtracting a particular fixed number to/from the previous
entry.
Growing Patterns:
Def: An arithmetic sequence begins with any number as the 1st entry,
but each subsequent entry is obtained by adding or subtracting a
particular fixed number to/from the previous entry.
Ex’s: 1
2
3
4
5
6 ….
Growing Patterns:
Def: An arithmetic sequence begins with any number as the 1st entry,
but each subsequent entry is obtained by adding or subtracting a
particular fixed number to/from the previous entry.
Ex’s: 1
7
2
3
4
5
11
15
19
23 ….
6 ….
Growing Patterns:
Def: An arithmetic sequence begins with any number as the 1st entry,
but each subsequent entry is obtained by adding or subtracting a
particular fixed number to/from the previous entry.
Ex’s: 1
2
3
4
5
7
11
15
19
23 ….
4
1
-2
-5 ….
6 ….
Arithmetic Sequences
• Question: How do we find the Nth entry in the sequence for
some whole number N?
Arithmetic Sequences
• Question: How do we find the Nth entry in the sequence for
some whole number N?
• See Activity 9Z
Arithmetic Sequences
In general, the Nth term of an arithmetic sequence is
(increase amount) x N + (0th entry)
where the 0th entry is found by subtracting the increase amount from
the 1st entry.
More Growing Patterns
• Def: A geometric sequence starts with any number as the 1st
entry, and then each subsequent entry is obtained by
multiplying or dividing by some fixed number. This fixed
number, when using multiplication, is called the ratio.
More Growing Patterns
• Def: A geometric sequence starts with any number as the 1st entry,
and then each subsequent entry is obtained by multiplying or dividing
by some fixed number. This fixed number, when using multiplication,
is called the ratio.
• Ex’s:
3
9
27
81 …..
More Growing Patterns
• Def: A geometric sequence starts with any number as the 1st entry,
and then each subsequent entry is obtained by multiplying or dividing
by some fixed number. This fixed number, when using multiplication,
is called the ratio.
• Ex’s:
3
9
27
81 …..
1
1/2
1/4
1/8 …..
Geometric Sequences
• Ex2: The amount of money in a savings account after N years of an
initial deposit of $1000 that earns 4% interest annually can be viewed
by the terms of the following geometric sequence:
1000
1040
1081.6
1124.87……
where the ratio of the sequence is 1.04. What is the value of the
account after 6 years?
Geometric Sequences
For a general geometric sequence, the Nth entry is
ratio
𝑁
∙ (0th entry)
where the 0th entry, if not given, can be found by dividing the 1st entry
by the ratio.
Other Sequences
• Fibonacci sequence: starts with 1 as the 1st and 2nd entries, with
each subsequent entry being found by adding its previous two entries
1 1 2 3 5 8 13 21 …..
Other Sequences
• Fibonacci sequence: starts with 1 as the 1st and 2nd entries, with
each subsequent entry being found by adding its previous two entries
1 1 2 3 5 8 13 21 …..
• Quadratic sequence: each entry is given by a quadratic expression
• Ex: 𝑁 2 + 2𝑁 − 3
0 5 12 21 32 …..
Other Sequences
• Fibonacci sequence: starts with 1 as the 1st and 2nd entries, with
each subsequent entry being found by adding its previous two entries
1 1 2 3 5 8 13 21 …..
• Quadratic sequence: each entry is given by a quadratic expression
• Ex: 𝑁 2 + 2𝑁 − 3
0 5 12 21 32 …..
• Non-repeating and non-growing:
3 1 4 1 5 9 2 6 5 3 5 9 …..