sequence introduction · sequences working with a sequence de ned by a formula: summation de nition...
TRANSCRIPT
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Sequence IntroductionSupplemental Material Not Found in You Text
Math 34 Fall 2016
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August 31, 2016
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Sequences
1 Sequence IntroDefinitionWorking with Sequences
2 Working with a Sequence Defined by a Formula:
3 SummationDefinitionExamples
4 Special Kinds of SequencesArithmetic SequencesGeometric Sequences
5 Homework
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Sequence Definition
A Sequence an ordered “list” of numbers.
Example Sequences:
1 2, 4, 6, 8, 10, . . .2 1
3 ,14 ,
15 ,
16 , . . .
3 1, 3, 9, 27, . . .
Since the list is ordered, we can associate each number inthe list with an integer (sometimes called the index)
The index helps you keep track of where in thesequence you are.Indices is the plural of index.
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Sequence Definition
A Sequence an ordered “list” of numbers.
Example Sequences:
1 2, 4, 6, 8, 10, . . .2 1
3 ,14 ,
15 ,
16 , . . .
3 1, 3, 9, 27, . . .
Since the list is ordered, we can associate each number inthe list with an integer (sometimes called the index)
The index helps you keep track of where in thesequence you are.Indices is the plural of index.
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Sequence Definition
A Sequence an ordered “list” of numbers.
Example Sequences:
1 2, 4, 6, 8, 10, . . .2 1
3 ,14 ,
15 ,
16 , . . .
3 1, 3, 9, 27, . . .
Since the list is ordered, we can associate each number inthe list with an integer (sometimes called the index)
The index helps you keep track of where in thesequence you are.Indices is the plural of index.
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Sequence Notation
We use the notation an for the number in the sequence atposition n.
So if we start counting the numbers in a sequence withindex 1 it looks like:
a1, a2, a3, a4, . . .
If we start counting the numbers in a sequence with index0 it looks like:
a0, a1, a2, a3, . . .
Although it is possible to start sequences with otherindices, in this course we will be primarily concerned withsequences that start with either 0 or 1.
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Sequence Notation
We use the notation an for the number in the sequence atposition n.
So if we start counting the numbers in a sequence withindex 1 it looks like:
a1, a2, a3, a4, . . .
If we start counting the numbers in a sequence with index0 it looks like:
a0, a1, a2, a3, . . .
Although it is possible to start sequences with otherindices, in this course we will be primarily concerned withsequences that start with either 0 or 1.
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Sequence Notation
We use the notation an for the number in the sequence atposition n.
So if we start counting the numbers in a sequence withindex 1 it looks like:
a1, a2, a3, a4, . . .
If we start counting the numbers in a sequence with index0 it looks like:
a0, a1, a2, a3, . . .
Although it is possible to start sequences with otherindices, in this course we will be primarily concerned withsequences that start with either 0 or 1.
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Working with Sequences
Consider the sequence
2, 4, 6, 8, 10, . . .
We can make a table for the sequence to help usunderstand how the index (order) relates to the numbersin the sequences.
Here we started the
indices with 1, sometimes
it will be more convent
to start with another
number such as 0.
Index Sequence(order) Value
1 a1 = 22 a2 = 43 a3 = 64 a4 = 85 a5 = 10...
...
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Working with Sequences
Consider the sequence
2, 4, 6, 8, 10, . . .
We can make a table for the sequence to help usunderstand how the index (order) relates to the numbersin the sequences.
Here we started the
indices with 1, sometimes
it will be more convent
to start with another
number such as 0.
Index Sequence(order) Value
1 a1 = 22 a2 = 43 a3 = 64 a4 = 85 a5 = 10...
...
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Working with Sequences (Cont.)
It’s redundant to in-clude the a1, a2, etcin the table, so we’llusually write tables asfollows:
Index Sequence(order) Value
1 22 43 64 85 10...
...
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Working with Sequences (Cont.)
For the Sequence2, 4, 6, 8, 10, . . .
Index Sequence(order) Value
1 22 43 64 85 10...
...
A What is the fourth term inthis sequence?
B What is the number atindex 4?
C What is the sixth term inthis sequence?
D Can you come up with aformula for the nth term inthis sequence?
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Can You Come Up With a Formula for theFollowing Sequence?
Index Sequence(order) Value
1 22 43 64 85 10...
...
The format should look like
an =(some formula involving n)
In other words, n is your variable
Another way to phrase the question:What formula gives you 2 as an output when the input is n = 1 ANDgives you 4 as an output when the input is n = 2 AND
gives you 6 as an output when the input is n = 3 ETC . . .
The output should be double the input so our formula isan = 2 · n
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Can You Come Up With a Formula for theFollowing Sequence?
Index Sequence(order) Value
1 22 43 64 85 10...
...
The format should look like
an =(some formula involving n)
In other words, n is your variable
Another way to phrase the question:What formula gives you 2 as an output when the input is n = 1 ANDgives you 4 as an output when the input is n = 2 AND
gives you 6 as an output when the input is n = 3 ETC . . .
The output should be double the input so our formula isan = 2 · n
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Can You Come Up With a Formula for theFollowing Sequence?
Index Sequence(order) Value
1 22 43 64 85 10...
...
The format should look like
an =(some formula involving n)
In other words, n is your variable
Another way to phrase the question:What formula gives you 2 as an output when the input is n = 1 ANDgives you 4 as an output when the input is n = 2 AND
gives you 6 as an output when the input is n = 3 ETC . . .
The output should be double the input so our formula isan = 2 · n
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Can You Come Up With a Formula for theFollowing Sequence? (Cont.)
Index Sequence(order) Value
n an = 2 · n1 a1 = 2 · 1 = 22 a2 = 2 · 2 = 43 a3 = 2 · 3 = 64 a4 = 2 · 4 = 85 a5 = 2 · 5 = 10...
...k ak = 2k
So we have two ways to represent the same sequence, either
2, 4, 6, 8, 10, . . . or an = 2·n where the index starts with 1
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Can You Come Up With a Formula for theFollowing Sequence? (Cont.)
Index Sequence(order) Value
n an = 2 · n1 a1 = 2 · 1 = 22 a2 = 2 · 2 = 43 a3 = 2 · 3 = 64 a4 = 2 · 4 = 85 a5 = 2 · 5 = 10...
...k ak = 2k
So we have two ways to represent the same sequence, either
2, 4, 6, 8, 10, . . . or an = 2·n where the index starts with 1
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Working with a Sequence Defined by a Formula:
Consider the sequence an = 4n + 3 where the index n startswith 1.
We can find the first term by subbing in n = 1:
a1 = 4 · 1 + 3 = 7
We can fill in a table to find the first 5 terms of thesequence:
Long Version Short VersionIndex Sequence(order) Value
n an = 4n + 3
1 a1 = 4 · 1 + 3 = 72 a2 = 4 · 2 + 3 = 113 a3 = 4 · 3 + 3 = 154 a4 = 4 · 4 + 3 = 195 a5 = 4 · 5 + 3 = 23...
...
Index Sequence(order) Value
n an = 4n + 3
1 72 113 154 195 23...
...
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Working with a Sequence Defined by a Formula:
Consider the sequence an = 4n + 3 where the index n startswith 1.
We can find the first term by subbing in n = 1:
a1 = 4 · 1 + 3 = 7
We can fill in a table to find the first 5 terms of thesequence:
Long Version Short VersionIndex Sequence(order) Value
n an = 4n + 3
1 a1 = 4 · 1 + 3 = 72 a2 = 4 · 2 + 3 = 113 a3 = 4 · 3 + 3 = 154 a4 = 4 · 4 + 3 = 195 a5 = 4 · 5 + 3 = 23...
...
Index Sequence(order) Value
n an = 4n + 3
1 72 113 154 195 23...
...
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Working with a Sequence Defined by a Formula:
Consider the sequence an = 4n + 3 where the index n startswith 1.
We can find the first term by subbing in n = 1:
a1 = 4 · 1 + 3 = 7
We can fill in a table to find the first 5 terms of thesequence:
Long Version Short VersionIndex Sequence(order) Value
n an = 4n + 3
1 a1 = 4 · 1 + 3 = 72 a2 = 4 · 2 + 3 = 113 a3 = 4 · 3 + 3 = 154 a4 = 4 · 4 + 3 = 195 a5 = 4 · 5 + 3 = 23...
...
Index Sequence(order) Value
n an = 4n + 3
1 72 113 154 195 23...
...
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Working with a Sequence Defined by a Formula:
Consider the sequence an = 4n + 3 where the index n startswith 1.
We can find the first term by subbing in n = 1:
a1 = 4 · 1 + 3 = 7
We can fill in a table to find the first 5 terms of thesequence:
Long Version Short VersionIndex Sequence(order) Value
n an = 4n + 3
1 a1 = 4 · 1 + 3 = 72 a2 = 4 · 2 + 3 = 113 a3 = 4 · 3 + 3 = 154 a4 = 4 · 4 + 3 = 195 a5 = 4 · 5 + 3 = 23...
...
Index Sequence(order) Value
n an = 4n + 3
1 72 113 154 195 23...
...
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Working with a Sequence Defined by a Formula:
Consider the sequence an = 4n + 3 where the index n startswith 1.
We can find the first term by subbing in n = 1:
a1 = 4 · 1 + 3 = 7
We can fill in a table to find the first 5 terms of thesequence:
Long Version Short VersionIndex Sequence(order) Value
n an = 4n + 3
1 a1 = 4 · 1 + 3 = 72 a2 = 4 · 2 + 3 = 113 a3 = 4 · 3 + 3 = 154 a4 = 4 · 4 + 3 = 195 a5 = 4 · 5 + 3 = 23...
...
Index Sequence(order) Value
n an = 4n + 3
1 72 113 154 195 23...
...
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Working with a Sequence Defined by a Formula:
Consider the sequence an = 4n + 3 where the index n startswith 1.
We can find the first term by subbing in n = 1:
a1 = 4 · 1 + 3 = 7
We can fill in a table to find the first 5 terms of thesequence:
Long Version Short VersionIndex Sequence(order) Value
n an = 4n + 3
1 a1 = 4 · 1 + 3 = 72 a2 = 4 · 2 + 3 = 113 a3 = 4 · 3 + 3 = 154 a4 = 4 · 4 + 3 = 195 a5 = 4 · 5 + 3 = 23...
...
Index Sequence(order) Value
n an = 4n + 3
1 72 113 154 195 23...
...
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Working with a Sequence Defined by a Formula(Cont.)
Consider the sequence an = 4n+ 3 where the index n starts with 1.
Since the index starts with 1:a1 is the first term, a2 is the second term, etc.
So we can use the formula for the sequence to find the10th term by finding a10:
The 10th term of the sequence is a10 = 4 · 10 + 3 = 43
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Working with a Sequence Defined by a Formula(Cont.)
Consider the sequence an = 4n+ 3 where the index n starts with 1.
Since the index starts with 1:a1 is the first term, a2 is the second term, etc.
So we can use the formula for the sequence to find the10th term by finding a10:
The 10th term of the sequence is a10 = 4 · 10 + 3 = 43
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Working with a Sequence Defined by a Formula(Cont.)
Consider the sequence an = 4n+ 3 where the index n starts with 1.
Since the index starts with 1:a1 is the first term, a2 is the second term, etc.
So we can use the formula for the sequence to find the10th term by finding a10:
The 10th term of the sequence is a10 = 4 · 10 + 3 = 43
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Working with a Sequence Defined by a Formula(Cont.)
Consider the sequence an = 4n+ 3
where the index n starts with 1.
Index Sequence(order) Value
n an = 4n + 3
1 72 113 154 195 23...
...
Questions
1 What is the third term in the sequence above?
2 What is the 7th term in the sequence above?
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Working with a Sequence Defined by a Formula(Part 2):
Consider the sequence an = 4n + 3 where the index n startswith 0.
We will see that this is a different sequence from theprevious example (even though the formula is the same).
Since the index starts with 0:a0 is the first term, a1 is the second term, etc.
We can find the first term.Since this sequence starts with n = 0, the first term is a0
a0 = 4 · 0 + 3 = 3
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Working with a Sequence Defined by a Formula(Part 2):
Consider the sequence an = 4n + 3 where the index n startswith 0.
We will see that this is a different sequence from theprevious example (even though the formula is the same).
Since the index starts with 0:a0 is the first term, a1 is the second term, etc.
We can find the first term.Since this sequence starts with n = 0, the first term is a0
a0 = 4 · 0 + 3 = 3
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Working with a Sequence Defined by a Formula(Part 2):
Consider the sequence an = 4n + 3 where the index n startswith 0.
We will see that this is a different sequence from theprevious example (even though the formula is the same).
Since the index starts with 0:a0 is the first term, a1 is the second term, etc.
We can find the first term.Since this sequence starts with n = 0, the first term is a0
a0 = 4 · 0 + 3 = 3
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Working with a Sequence Defined by a Formula(Part 2):
Consider the sequence an = 4n + 3 where the index n startswith 0.
We will see that this is a different sequence from theprevious example (even though the formula is the same).
Since the index starts with 0:a0 is the first term, a1 is the second term, etc.
We can find the first term.Since this sequence starts with n = 0, the first term is a0
a0 = 4 · 0 + 3 = 3
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Working with a Sequence Defined by a Formula(Part 2):
Consider the sequence an = 4n + 3 where the index n startswith 0.
We can fill in a table to find the first five terms of thesequence:
Long Version Short VersionIndex Sequence(order) Value
n an = 4n + 3
0 a0 = 4 · 0 + 3 = 3
1 a1 = 4 · 1 + 3 = 72 a2 = 4 · 2 + 3 = 113 a3 = 4 · 3 + 3 = 154 a4 = 4 · 4 + 3 = 19...
...
Index Sequence(order) Value
n an = 4n + 3
0 31 72 113 154 19...
...
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Working with a Sequence Defined by a Formula(Part 2):
Consider the sequence an = 4n + 3 where the index n startswith 0.
We can fill in a table to find the first five terms of thesequence:
Long Version Short VersionIndex Sequence(order) Value
n an = 4n + 3
0 a0 = 4 · 0 + 3 = 31 a1 = 4 · 1 + 3 = 7
2 a2 = 4 · 2 + 3 = 113 a3 = 4 · 3 + 3 = 154 a4 = 4 · 4 + 3 = 19...
...
Index Sequence(order) Value
n an = 4n + 3
0 31 72 113 154 19...
...
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Working with a Sequence Defined by a Formula(Part 2):
Consider the sequence an = 4n + 3 where the index n startswith 0.
We can fill in a table to find the first five terms of thesequence:
Long Version Short VersionIndex Sequence(order) Value
n an = 4n + 3
0 a0 = 4 · 0 + 3 = 31 a1 = 4 · 1 + 3 = 72 a2 = 4 · 2 + 3 = 113 a3 = 4 · 3 + 3 = 154 a4 = 4 · 4 + 3 = 19...
...
Index Sequence(order) Value
n an = 4n + 3
0 31 72 113 154 19...
...
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Working with a Sequence Defined by a Formula(Part 2):
Consider the sequence an = 4n + 3 where the index n startswith 0.
We can fill in a table to find the first five terms of thesequence:
Long Version Short VersionIndex Sequence(order) Value
n an = 4n + 3
0 a0 = 4 · 0 + 3 = 31 a1 = 4 · 1 + 3 = 72 a2 = 4 · 2 + 3 = 113 a3 = 4 · 3 + 3 = 154 a4 = 4 · 4 + 3 = 19...
...
Index Sequence(order) Value
n an = 4n + 3
0 31 72 113 154 19...
...
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Working with a Sequence Defined by a Formula(Part 2):
Consider the sequence an = 4n + 3 where the index n startswith 0.
So we can use the formula for the sequence to find the10th.
Since the index starts with 0, the 10th term of thesequences is a9:The 10th term of the sequence is a9 = 4 · 9 + 3 = 39
Questions
1 What is the third term in the sequence above?
2 What is the 7th term in the sequence above?
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Working with a Sequence Defined by a Formula(Part 2):
Consider the sequence an = 4n + 3 where the index n startswith 0.
So we can use the formula for the sequence to find the10th.Since the index starts with 0, the 10th term of thesequences is a9:The 10th term of the sequence is a9 = 4 · 9 + 3 = 39
Questions
1 What is the third term in the sequence above?
2 What is the 7th term in the sequence above?
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Working with a Sequence Defined by a Formula(Part 2):
Consider the sequence an = 4n + 3 where the index n startswith 0.
So we can use the formula for the sequence to find the10th.Since the index starts with 0, the 10th term of thesequences is a9:The 10th term of the sequence is a9 = 4 · 9 + 3 = 39
Questions
1 What is the third term in the sequence above?
2 What is the 7th term in the sequence above?
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Key Points about indices
If the sequence starts with n = 1
The first term of the sequence is a1
The second term of the sequence is a2
The third term of the sequence is a3
etc...
If the sequence starts with n = 0
The first term of the sequence is a0
The second term of the sequence is a1
The third term of the sequence is a2
etc...
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Summation (Definition)
Once we start using sequences for our real world exampleswe’ll often want add up the first several terms of asequence.
The nth Partial Sum of the Sequence an is the sum ofthe first n terms of the sequence. We use Sn to denote thenth Partial Sum.
If the sequence starts with n = 1:
Sn = a1 + a2 + s3 + · · · + an
If the sequence starts with n = 0:
Sn = a0 + a1 + a2 + · · · + an−1
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Summation (Definition)
Once we start using sequences for our real world exampleswe’ll often want add up the first several terms of asequence.
The nth Partial Sum of the Sequence an is the sum ofthe first n terms of the sequence. We use Sn to denote thenth Partial Sum.
If the sequence starts with n = 1:
Sn = a1 + a2 + s3 + · · · + an
If the sequence starts with n = 0:
Sn = a0 + a1 + a2 + · · · + an−1
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Summation (Definition)
Once we start using sequences for our real world exampleswe’ll often want add up the first several terms of asequence.
The nth Partial Sum of the Sequence an is the sum ofthe first n terms of the sequence. We use Sn to denote thenth Partial Sum.
If the sequence starts with n = 1:
Sn = a1 + a2 + s3 + · · · + an
If the sequence starts with n = 0:
Sn = a0 + a1 + a2 + · · · + an−1
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Summation Examples
First Example: For the sequence an = 4 · n + 3 startingwith n = 1, find both the third partial sum and the fifthpartial sum.
Recall... Since the sequence starts w/ n = 1:Index Sequence(order) Value
n an = 4n + 3
1 72 113 154 195 23...
...
The third partial sum is:S3 = a1 + a2 + a3
S3 = 7 + 11 + 15 = 33
The fifth partial sum is:S5 = a1 + a2 + a3 + a4 + a5
S5 = 7 + 11 + 15 + 19 + 23 = 75
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Summation Examples
First Example: For the sequence an = 4 · n + 3 startingwith n = 1, find both the third partial sum and the fifthpartial sum.
Recall... Since the sequence starts w/ n = 1:Index Sequence(order) Value
n an = 4n + 3
1 72 113 154 195 23...
...
The third partial sum is:S3 = a1 + a2 + a3
S3 = 7 + 11 + 15 = 33
The fifth partial sum is:S5 = a1 + a2 + a3 + a4 + a5
S5 = 7 + 11 + 15 + 19 + 23 = 75
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Summation Examples
First Example: For the sequence an = 4 · n + 3 startingwith n = 1, find both the third partial sum and the fifthpartial sum.
Recall... Since the sequence starts w/ n = 1:Index Sequence(order) Value
n an = 4n + 3
1 72 113 154 195 23...
...
The third partial sum is:S3 = a1 + a2 + a3
S3 = 7 + 11 + 15 = 33
The fifth partial sum is:S5 = a1 + a2 + a3 + a4 + a5
S5 = 7 + 11 + 15 + 19 + 23 = 75
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Summation Examples
First Example: For the sequence an = 4 · n + 3 startingwith n = 1, find both the third partial sum and the fifthpartial sum.
Recall... Since the sequence starts w/ n = 1:Index Sequence(order) Value
n an = 4n + 3
1 72 113 154 195 23...
...
The third partial sum is:S3 = a1 + a2 + a3
S3 = 7 + 11 + 15 = 33
The fifth partial sum is:S5 = a1 + a2 + a3 + a4 + a5
S5 = 7 + 11 + 15 + 19 + 23 = 75
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Summation Examples
First Example: For the sequence an = 4 · n + 3 startingwith n = 1, find both the third partial sum and the fifthpartial sum.
Recall... Since the sequence starts w/ n = 1:Index Sequence(order) Value
n an = 4n + 3
1 72 113 154 195 23...
...
The third partial sum is:S3 = a1 + a2 + a3
S3 = 7 + 11 + 15 = 33
The fifth partial sum is:S5 = a1 + a2 + a3 + a4 + a5
S5 = 7 + 11 + 15 + 19 + 23 = 75
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Summation Examples
First Example: For the sequence an = 4 · n + 3 startingwith n = 1, find both the third partial sum and the fifthpartial sum.
Recall... Since the sequence starts w/ n = 1:Index Sequence(order) Value
n an = 4n + 3
1 72 113 154 195 23...
...
The third partial sum is:S3 = a1 + a2 + a3
S3 = 7 + 11 + 15 = 33
The fifth partial sum is:S5 = a1 + a2 + a3 + a4 + a5
S5 = 7 + 11 + 15 + 19 + 23 = 75
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Summation Examples (Cont.)
Second Example: For the sequence an = 4 · n + 3starting with n = 0, find both the third partial sum andthe fifth partial sum.
Recall... Since the sequence starts w/ n = 0:Index Sequence(order) Value
n an = 4n + 3
0 31 72 113 154 19...
...
The third partial sum is:S3 = a0 + a1 + a2
S3 = 3 + 7 + 11 = 21
The fifth partial sum is:S5 = a0 + a1 + a2 + a3 + a4
S5 = 3 + 7 + 11 + 15 + 19 = 55
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Summation Examples (Cont.)
Second Example: For the sequence an = 4 · n + 3starting with n = 0, find both the third partial sum andthe fifth partial sum.
Recall... Since the sequence starts w/ n = 0:Index Sequence(order) Value
n an = 4n + 3
0 31 72 113 154 19...
...
The third partial sum is:S3 = a0 + a1 + a2
S3 = 3 + 7 + 11 = 21
The fifth partial sum is:S5 = a0 + a1 + a2 + a3 + a4
S5 = 3 + 7 + 11 + 15 + 19 = 55
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Summation Examples (Cont.)
Second Example: For the sequence an = 4 · n + 3starting with n = 0, find both the third partial sum andthe fifth partial sum.
Recall... Since the sequence starts w/ n = 0:Index Sequence(order) Value
n an = 4n + 3
0 31 72 113 154 19...
...
The third partial sum is:S3 = a0 + a1 + a2
S3 = 3 + 7 + 11 = 21
The fifth partial sum is:S5 = a0 + a1 + a2 + a3 + a4
S5 = 3 + 7 + 11 + 15 + 19 = 55
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Summation Examples (Cont.)
Second Example: For the sequence an = 4 · n + 3starting with n = 0, find both the third partial sum andthe fifth partial sum.
Recall... Since the sequence starts w/ n = 0:Index Sequence(order) Value
n an = 4n + 3
0 31 72 113 154 19...
...
The third partial sum is:S3 = a0 + a1 + a2
S3 = 3 + 7 + 11 = 21
The fifth partial sum is:S5 = a0 + a1 + a2 + a3 + a4
S5 = 3 + 7 + 11 + 15 + 19 = 55
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Summation Examples (Cont.)
Second Example: For the sequence an = 4 · n + 3starting with n = 0, find both the third partial sum andthe fifth partial sum.
Recall... Since the sequence starts w/ n = 0:Index Sequence(order) Value
n an = 4n + 3
0 31 72 113 154 19...
...
The third partial sum is:S3 = a0 + a1 + a2
S3 = 3 + 7 + 11 = 21
The fifth partial sum is:S5 = a0 + a1 + a2 + a3 + a4
S5 = 3 + 7 + 11 + 15 + 19 = 55
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Summation Examples (Cont.)
Second Example: For the sequence an = 4 · n + 3starting with n = 0, find both the third partial sum andthe fifth partial sum.
Recall... Since the sequence starts w/ n = 0:Index Sequence(order) Value
n an = 4n + 3
0 31 72 113 154 19...
...
The third partial sum is:S3 = a0 + a1 + a2
S3 = 3 + 7 + 11 = 21
The fifth partial sum is:S5 = a0 + a1 + a2 + a3 + a4
S5 = 3 + 7 + 11 + 15 + 19 = 55
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Summation Examples (Cont.)
A Word of Caution:
For these examples we had already filled out the table.
In some questions you may have to do that yourself as thefirst step in finding a partial sum.
(Eventually we’ll have a shortcut formula for certainpartial sums)
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Summation Examples (Cont.)
A Word of Caution:
For these examples we had already filled out the table.
In some questions you may have to do that yourself as thefirst step in finding a partial sum.
(Eventually we’ll have a shortcut formula for certainpartial sums)
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Two Special Kinds of Sequences
An Arithmetic Sequence is a sequence where thedifference between any two consecutive numbers in thesequence is constant.
In other words: ak+1 − ak = d where d is a constant.
A Geometric Sequence is a sequence where the ratiobetween any two consecutive numbers in the sequence is aconstant.
In other words: ak+1/ak = r where r is a constant.
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Two Special Kinds of Sequences
An Arithmetic Sequence is a sequence where thedifference between any two consecutive numbers in thesequence is constant.
In other words: ak+1 − ak = d where d is a constant.
A Geometric Sequence is a sequence where the ratiobetween any two consecutive numbers in the sequence is aconstant.
In other words: ak+1/ak = r where r is a constant.
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Two Special Kinds of Sequences
An Arithmetic Sequence is a sequence where thedifference between any two consecutive numbers in thesequence is constant.
In other words: ak+1 − ak = d where d is a constant.
A Geometric Sequence is a sequence where the ratiobetween any two consecutive numbers in the sequence is aconstant.
In other words: ak+1/ak = r where r is a constant.
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Arithmetic Sequences
Example of an Arithmetic Sequence: 3, 5, 7, 9, 11, . . .
3︸︷︷︸a1
, 5︸︷︷︸a2
, 7︸︷︷︸a3
, 9︸︷︷︸a4
, 11︸︷︷︸a5
, . . .
We’ll check for the constant difference:
(k=1) a2 − a1 = 5 − 3 = 2(k=2) a3 − a2 = 7 − 5 = 2(k=3) a4 − a3 = 9 − 7 = 2(k=4) a5 − a4 = 11 − 9 = 2
The constant difference d = 2
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Arithmetic Sequences
Example of an Arithmetic Sequence: 3, 5, 7, 9, 11, . . .
3︸︷︷︸a1
, 5︸︷︷︸a2
, 7︸︷︷︸a3
, 9︸︷︷︸a4
, 11︸︷︷︸a5
, . . .
We’ll check for the constant difference:
(k=1)
a2 − a1 = 5 − 3 = 2(k=2) a3 − a2 = 7 − 5 = 2(k=3) a4 − a3 = 9 − 7 = 2(k=4) a5 − a4 = 11 − 9 = 2
The constant difference d = 2
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Arithmetic Sequences
Example of an Arithmetic Sequence: 3, 5, 7, 9, 11, . . .
3︸︷︷︸a1
, 5︸︷︷︸a2
, 7︸︷︷︸a3
, 9︸︷︷︸a4
, 11︸︷︷︸a5
, . . .
We’ll check for the constant difference:
(k=1) a2 − a1
= 5 − 3 = 2(k=2) a3 − a2 = 7 − 5 = 2(k=3) a4 − a3 = 9 − 7 = 2(k=4) a5 − a4 = 11 − 9 = 2
The constant difference d = 2
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Arithmetic Sequences
Example of an Arithmetic Sequence: 3, 5, 7, 9, 11, . . .
3︸︷︷︸a1
, 5︸︷︷︸a2
, 7︸︷︷︸a3
, 9︸︷︷︸a4
, 11︸︷︷︸a5
, . . .
We’ll check for the constant difference:
(k=1) a2 − a1 = 5 − 3 = 2
(k=2) a3 − a2 = 7 − 5 = 2(k=3) a4 − a3 = 9 − 7 = 2(k=4) a5 − a4 = 11 − 9 = 2
The constant difference d = 2
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Arithmetic Sequences
Example of an Arithmetic Sequence: 3, 5, 7, 9, 11, . . .
3︸︷︷︸a1
, 5︸︷︷︸a2
, 7︸︷︷︸a3
, 9︸︷︷︸a4
, 11︸︷︷︸a5
, . . .
We’ll check for the constant difference:
(k=1) a2 − a1 = 5 − 3 = 2(k=2) a3 − a2 = 7 − 5 = 2
(k=3) a4 − a3 = 9 − 7 = 2(k=4) a5 − a4 = 11 − 9 = 2
The constant difference d = 2
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Arithmetic Sequences
Example of an Arithmetic Sequence: 3, 5, 7, 9, 11, . . .
3︸︷︷︸a1
, 5︸︷︷︸a2
, 7︸︷︷︸a3
, 9︸︷︷︸a4
, 11︸︷︷︸a5
, . . .
We’ll check for the constant difference:
(k=1) a2 − a1 = 5 − 3 = 2(k=2) a3 − a2 = 7 − 5 = 2(k=3) a4 − a3 = 9 − 7 = 2(k=4) a5 − a4 = 11 − 9 = 2
The constant difference d = 2
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Arithmetic Sequences
Example of an Arithmetic Sequence: 3, 5, 7, 9, 11, . . .
3︸︷︷︸a1
, 5︸︷︷︸a2
, 7︸︷︷︸a3
, 9︸︷︷︸a4
, 11︸︷︷︸a5
, . . .
We’ll check for the constant difference:
(k=1) a2 − a1 = 5 − 3 = 2(k=2) a3 − a2 = 7 − 5 = 2(k=3) a4 − a3 = 9 − 7 = 2(k=4) a5 − a4 = 11 − 9 = 2
The constant difference d = 2
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Geometric Sequences
Example of an Geometric Sequence: 2, 6, 18, 54, 162 . . .
2︸︷︷︸a1
, 6︸︷︷︸a2
, 18︸︷︷︸a3
, 54︸︷︷︸a4
, 162︸︷︷︸a5
, . . .
We’ll check for the constant ratio:
(k=1) a2/a1 = 6/2 = 3(k=2) a3/a2 = 18/6 = 3(k=3) a4/a3 = 54/18 = 3(k=4) a5/a4 = 162/54 = 3
The constant ratio r = 3
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Geometric Sequences
Example of an Geometric Sequence: 2, 6, 18, 54, 162 . . .
2︸︷︷︸a1
, 6︸︷︷︸a2
, 18︸︷︷︸a3
, 54︸︷︷︸a4
, 162︸︷︷︸a5
, . . .
We’ll check for the constant ratio:
(k=1)
a2/a1 = 6/2 = 3(k=2) a3/a2 = 18/6 = 3(k=3) a4/a3 = 54/18 = 3(k=4) a5/a4 = 162/54 = 3
The constant ratio r = 3
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Geometric Sequences
Example of an Geometric Sequence: 2, 6, 18, 54, 162 . . .
2︸︷︷︸a1
, 6︸︷︷︸a2
, 18︸︷︷︸a3
, 54︸︷︷︸a4
, 162︸︷︷︸a5
, . . .
We’ll check for the constant ratio:
(k=1) a2/a1
= 6/2 = 3(k=2) a3/a2 = 18/6 = 3(k=3) a4/a3 = 54/18 = 3(k=4) a5/a4 = 162/54 = 3
The constant ratio r = 3
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Geometric Sequences
Example of an Geometric Sequence: 2, 6, 18, 54, 162 . . .
2︸︷︷︸a1
, 6︸︷︷︸a2
, 18︸︷︷︸a3
, 54︸︷︷︸a4
, 162︸︷︷︸a5
, . . .
We’ll check for the constant ratio:
(k=1) a2/a1 = 6/2 = 3
(k=2) a3/a2 = 18/6 = 3(k=3) a4/a3 = 54/18 = 3(k=4) a5/a4 = 162/54 = 3
The constant ratio r = 3
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Geometric Sequences
Example of an Geometric Sequence: 2, 6, 18, 54, 162 . . .
2︸︷︷︸a1
, 6︸︷︷︸a2
, 18︸︷︷︸a3
, 54︸︷︷︸a4
, 162︸︷︷︸a5
, . . .
We’ll check for the constant ratio:
(k=1) a2/a1 = 6/2 = 3(k=2) a3/a2 = 18/6 = 3
(k=3) a4/a3 = 54/18 = 3(k=4) a5/a4 = 162/54 = 3
The constant ratio r = 3
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Geometric Sequences
Example of an Geometric Sequence: 2, 6, 18, 54, 162 . . .
2︸︷︷︸a1
, 6︸︷︷︸a2
, 18︸︷︷︸a3
, 54︸︷︷︸a4
, 162︸︷︷︸a5
, . . .
We’ll check for the constant ratio:
(k=1) a2/a1 = 6/2 = 3(k=2) a3/a2 = 18/6 = 3(k=3) a4/a3 = 54/18 = 3(k=4) a5/a4 = 162/54 = 3
The constant ratio r = 3
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Geometric Sequences
Example of an Geometric Sequence: 2, 6, 18, 54, 162 . . .
2︸︷︷︸a1
, 6︸︷︷︸a2
, 18︸︷︷︸a3
, 54︸︷︷︸a4
, 162︸︷︷︸a5
, . . .
We’ll check for the constant ratio:
(k=1) a2/a1 = 6/2 = 3(k=2) a3/a2 = 18/6 = 3(k=3) a4/a3 = 54/18 = 3(k=4) a5/a4 = 162/54 = 3
The constant ratio r = 3
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Determining if a sequence is Arithmetic,Geometric, or Neither:
You’ll have to check (separately) if the sequence isArithmetic and if it is Geometric.
If the sequence is given with a formula, you’ll need towrite out the first 4 or 5 terms of the sequence.
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Determining if a sequence is Arithmetic,Geometric, or Neither (Cont.)
Determine if the following sequences are Arithmetic,Geometric, or Neither
1 20, 10, 5, 2.5, 1.25, . . .
2 1, 4, 9, 25, 36, . . .
3 2, 9, 16, 23, . . .
4 an = 7 + 3n starting at index n = 0
5 an = 2n starting at index n = 1
SequenceIntroduction
Math 34 Fall2016
SequenceIntro
Definition
Working withSequences
Working witha SequenceDefined by aFormula:
Summation
Definition
Examples
Special Kindsof Sequences
ArithmeticSequences
GeometricSequences
Homework
Homework
It is NOT in your book.
It IS at the end of the printout on the course website.