Transcript Sequence

Math Journal 9-4
Evaluate
1. −37 − −6 =
Simplify
2
2
2. 12𝑦 − 3𝑦 − 15𝑦 + 14 𝑦
4. Evaluate 𝒇 𝒙 = 𝟓𝒙 − 𝟏𝟐
When 𝒙 = 8 and when 𝒙 = -5
3.
Is this graph a Function?
Yes or No
Domain: ______________________
Range: _______________________
Unit 2 Day 3: Types
of Sequences
Essential Questions: How can we describe two types of
sequences? How can the Fibonacci sequence be used
to explain a recursive pattern?
Vocabulary
• Sequence: an ordered list of numbers that often forms a pattern.
• Term: each number in a sequence (synonyms – member, element)
• Arithmetic Sequence: a sequence formed by adding a constant
number to each previous term. This constant number is also called
the common difference.
• Recursive Sequence: a sequence which uses the previous term to
determine the next term.
Describe the pattern, then find the next two
numbers.
2, 5, 8, 11
+3 +3 +3
Look at your vocab from today!
What type of sequence would this
be?
Arithmetic!! We are adding the same
number, 3, each time.
The “Common Difference” is 3.
The pattern is “add 3 to the previous term”. To
find the next two numbers, you add 3 to each
previous term. 11 +3 = 14, and 14 + 3 = 17.
Example 1: Find the common
difference for the arithmetic sequence:
-7, -11, -15, -19
-11 - (-7)
-11 + 7
d = -4
Example 2
2, ___, ___, ___, 30
What is the total
difference between 2
and 30?
30 - 2 = 28
+?
+?
+?
+7
+7
+7
+?
+7
These numbers form an
arithmetic sequence.
What is the common
difference? What are the
three missing numbers?
If we have to split that
total difference
between all four
arrows, what would the
“common difference”
be?
28/4 = 7
Analyzing Example 2:
Complete the table. What is the second term in the sequence?
Term #
Term
Common
Difference
1
2
+7
9
2
9
+7
16
3
16
+7
23
4
23
+7
30
5
30
+7
37
Answer
Example 3
Buses on your route run every 7 minutes between
6:30 A.M. and 9:00 A.M. You get to the bus stop at
7:07 A.M. Use the information to complete the table,
then determine how long will you have to wait for a
bus.
Bus Stop #
Arriving Time
Traveling Time
Next Stop
1
6:30 AM
7 Min
6:37 AM
Analyzing Example 3:
Bus #
Arriving Time
Traveling
Time
1
6:30 AM
7 Min
6:37 AM
2
6:37 AM
7 Min
6:44 AM
3
6:44 AM
7 Min
6:51 AM
4
6:51 AM
7 Min
6:58 AM
5
6:58 AM
7 Min
7:05 AM
6
7:05 AM
7 Min
7:12 AM
7
7:12 AM
7 Min
7:19 AM
Next Stop
How long do you have to wait for the
next bus if you arrive at 7:07 AM?
Quick Write:
Look at the two tables you have in
your notes… by definition of a
function, do you believe a
sequence is a function? Why or
why not?
Recursive Sequences
}+
Not all patterns are arithmetic sequences. Some
sequences are recursive (requires a previous terms to
continue).
For example, if 1 and 1 were the first two terms in a
sequence and the function rule was to add two previous
terms to find the next, what would be the next number?
2
}+
1 1
1 1 2
3
}+
The Fibonacci Numbers
1 1 2 3
5
8 13 21 34 55
This specific sequence is called the Fibonacci
Sequence. It is a natural pattern that exists and
shows how a recursive pattern may appear in
numbers.
Example 4: Complete the recursive sequence table
using the same rule as the Fibonacci sequence.
Term Number
1
2
3
4
5
6
7
Term
3
7
10
17
27
44
71
What is the 7th term in the sequence?
Fibonacci Numbers In Nature
•
•
The Fibonacci numbers are found many places
in the natural world, including:
• The number of flower petals.
• The branching behavior of plants.
• The growth patterns of sunflowers and
pinecones, ……
It is believed that the spiral nature of plant
growth accounts for this phenomenon.
Fibonacci Numbers In Anatomy
The lengths of
bones in a hand
are Fibonacci
numbers.
http://dsc.discovery.com/tv-shows/othershows/videos/assignment-discoveryfibonacci-sequence.htm
Summary
Essential Questions: How can we describe two types of
sequences? How can the Fibonacci sequence be used to
explain a recursive pattern?
Take 1 minute to write 2 sentences answering the
essential questions.