Sequences & Summation Sequences & Summation NotationNotation

8.18.1

JMerrill, 2007JMerrill, 2007

Revised 2008Revised 2008

Sequences In Elementary School…Sequences In Elementary School…

1212

32

And…And…

17

12

EvenEven

12

22

SequencesSequences

SEQUENCE SEQUENCE - a set of numbers, called - a set of numbers, called terms, arranged in a particular order. terms, arranged in a particular order.

SequencesSequences

An An infiniteinfinite sequence is a function whose sequence is a function whose domain is the set of positive integers. The domain is the set of positive integers. The function values afunction values a11, a, a22, a, a33, …, a, …, ann… are the … are the

terms of the sequence.terms of the sequence.

If the domain of the sequence consists of If the domain of the sequence consists of the first the first nn positive integers only, the positive integers only, the sequence is a sequence is a finitefinite sequence. sequence.

n is the term number.

ExamplesExamples

Finite sequence: Finite sequence: 2, 6, 10, 142, 6, 10, 14

Infinite sequence:Infinite sequence:1 1 1 1

, , , ,...2 4 8 16

Writing the Terms of a SequenceWriting the Terms of a Sequence

Write the first 4 terms of the sequence Write the first 4 terms of the sequence a ann = 3n – 2 = 3n – 2

aa11 = 3(1) – 2 = 1 = 3(1) – 2 = 1

aa22 = 3(2) – 2 = 4 = 3(2) – 2 = 4

aa33 = 3(3) – 2 = 7 = 3(3) – 2 = 7

aa44 = 3(4) – 2 = 10 = 3(4) – 2 = 10

Calculator steps in LIST

Writing the Terms of a SequenceWriting the Terms of a Sequence

Write the first 4 terms of the sequence Write the first 4 terms of the sequence a ann = 3 + (-1) = 3 + (-1)nn

aa11 = 3 + (-1) = 3 + (-1)1 1 == 22

aa22 = 3 + (-1) = 3 + (-1)2 2 == 44

aa33 = 3 + (-1) = 3 + (-1)3 3 == 22

aa44 = 3 + (-1) = 3 + (-1)4 4 == 44

You DoYou Do

Write the first 4 terms of the sequence Write the first 4 terms of the sequence

n

n

( 1)a

2n 1

1 1 1 11, , , ,

3 5 7 9

GraphsGraphs

Consider the infinite Consider the infinite sequencesequence

Because a sequence is a Because a sequence is a function whose domain is function whose domain is the set of positive the set of positive integers, the graph of a integers, the graph of a sequence is a set of sequence is a set of distinct points.distinct points.

The first term is ½ , the The first term is ½ , the 22ndnd term is ¼ … term is ¼ …So, the ordered pairs are So, the ordered pairs are (1, ½ ), (2, ¼ )…(1, ½ ), (2, ¼ )…

1 1 1 1 1, , , ,..., ...

2 4 8 16 2

n

Finding the nFinding the nthth Term of a Sequence Term of a Sequence

Write an expression for the nWrite an expression for the nthth term (a term (ann) of ) of

the sequence 1, 3, 5, 7…the sequence 1, 3, 5, 7…

n: 1, 2, 3, 4…nn: 1, 2, 3, 4…n

Terms: 1, 3, 5, 7…aTerms: 1, 3, 5, 7…ann

Apparent pattern: each term is 1 less than Apparent pattern: each term is 1 less than twice n. So, the apparent ntwice n. So, the apparent nthth term is term is

aann = 2n - 1 = 2n - 1

Always compare the term to the term number

Finding the nFinding the nthth Term of a Sequence Term of a SequenceYou DoYou Do

Write an expression for the nWrite an expression for the nthth term (a term (ann) of ) of

the sequence the sequence

Apparent pattern:Apparent pattern:The numerator is 1; the denominator is the square of n.

n = 1, 2, 3, 4…n1 1 1 11, , , , ...

4 9 16 25 1 1 1 11, , , , ...

4 9 16 25 nTerms a

2

1na

n

Recursive DefinitionRecursive Definition

Sometimes a sequence is defined by Sometimes a sequence is defined by giving the value of agiving the value of ann in terms of the in terms of the preceding term, apreceding term, an-1n-1. . A recursive A recursive sequence consists of 2 parts:sequence consists of 2 parts:An An initial conditioninitial condition that tells where the that tells where the sequence starts.sequence starts.A A recursive equationrecursive equation (or formula) that tells (or formula) that tells how many terms in the sequence are how many terms in the sequence are related to the preceding term.related to the preceding term.

ExampleExample

If aIf ann = = aan-1n-1 + 4 and + 4 and aa11 = 3, give the first five = 3, give the first five

terms of the sequence.terms of the sequence.

aa11 = 3 = 3

If n = 2: If n = 2: aa22 = = aa11 + 4 = 3 + 4 = 7+ 4 = 3 + 4 = 7

If n = 3: If n = 3: aa33 = = aa22 + 4 = 7 + 4 = 11+ 4 = 7 + 4 = 11

If n = 4: If n = 4: aa44 = = aa33 + 4 = 11 + 4 = 15+ 4 = 11 + 4 = 15

If n = 5: If n = 5: aa5 = = aa44 + + 4 = 15 + 4 = 19 4 = 15 + 4 = 19

A Famous Recursive SequenceA Famous Recursive Sequence

The Fibonacci Sequence is very well The Fibonacci Sequence is very well known because it appears in nature.known because it appears in nature.

The sequence is 1, 1, 2, 3, 5, 8, 13…The sequence is 1, 1, 2, 3, 5, 8, 13…

Apparent pattern?Apparent pattern?

Each term is the sum of the preceding 2 Each term is the sum of the preceding 2 termsterms

The nth term isThe nth term is

aann = a = an-2n-2 + a + an-1n-1

ExampleExample

Write the first 4 terms of the sequence Write the first 4 terms of the sequence

aa00 = 1 = 1

aa11 = 2 = 2

aa22 = 2 = 2

aa33 = 4/3 = 4/3

aa44 = 2/3 = 2/3

n

n

2a , begin with n 0

n!

Factorial NotationFactorial Notation

Products of consecutive positive integers Products of consecutive positive integers occur quite often in sequences. These occur quite often in sequences. These products can be expressed in factorial products can be expressed in factorial notation:notation:1! = 11! = 12! = 2 2! = 2 ● 1 = 2● 1 = 23! = 3 ●2 ●1 = 63! = 3 ●2 ●1 = 64! = 4 ●3 ●2 ●1 = 244! = 4 ●3 ●2 ●1 = 245! = 5 ●4 ●3 ●2 ●1 = 1205! = 5 ●4 ●3 ●2 ●1 = 120

The factorial key can be found in MATH PRB:4 on your calculator

0!, by definition, = 1

ExampleExample

Write the first four terms of the sequence Write the first four terms of the sequence

n

n

2a

(n 1)!

1

1

2

2

3

3

4

4

2 2 2a 2

(1 1)! 0! 1

2 4 4a 4

(2 1)! 1! 1

2 8 8a 4

(3 1)! 2! 2

2 16 16 8a

(4 1)! 3! 6 3

Evaluating Factorials in FractionsEvaluating Factorials in Fractions

Evaluate:Evaluate:

n 1 !10!

2!8! n!

10 9 8! 9045

2 1 8! 2

(n 1) n!n 1

n!

DefinitionsDefinitions

The words sequences and series are often The words sequences and series are often used interchangeably in everyday used interchangeably in everyday conversation. (A person may refer to a conversation. (A person may refer to a sequence of events or a series of events.) sequence of events or a series of events.) In mathematics, they are very different.In mathematics, they are very different.

Sequence:Sequence: a set of numbers, terms, a set of numbers, terms, arranged in a particular orderarranged in a particular order

Series:Series: the sum of a sequence the sum of a sequence

ExamplesExamples

Finite sequence: Finite sequence: 2, 6, 10, 142, 6, 10, 14

Finite series: Finite series: 2 + 6 + 10 + 142 + 6 + 10 + 14

Infinite sequence:Infinite sequence:

Infinite series: Infinite series:

1 1 1 1, , , ,...

2 4 8 16

1 1 1 1...

2 4 8 16

Intro to SigmaIntro to Sigma

The Greek letter (sigma) is often used The Greek letter (sigma) is often used in mathematics to represent a sum (series) in mathematics to represent a sum (series) in abbreviated form.in abbreviated form.

Example:Example: which which

can be read as “the sum of kcan be read as “the sum of k22 for values of for values of k from 1 to 100.” k from 1 to 100.”

1002 2 2 2 2

1

1 2 3 ... 100k

k

Definition of a SeriesDefinition of a Series

Consider the infinite series aConsider the infinite series a11, a, a22, … a, … ann……

The sum of the first n terms is a finite The sum of the first n terms is a finite series (or partial sum) and is denoted byseries (or partial sum) and is denoted by

The sum of all terms of an infinite The sum of all terms of an infinite sequence is called an infinite series and is sequence is called an infinite series and is denoted by denoted by

1

n

ii

a

1i

i

a

Sigma ContinuedSigma Continued

Similarly, the symbol is read “theSimilarly, the symbol is read “the

sum of 3k for values of k from 5 to 10.”sum of 3k for values of k from 5 to 10.”

This means that the symbol represents the This means that the symbol represents the series whose terms are obtained by series whose terms are obtained by evaluating 3k for k = 5, k = 6, and so on, to evaluating 3k for k = 5, k = 6, and so on, to k = 10. k = 10.

10

5

3k

k

10

5

3 3(5) 3(6) 3(7) 3(8) 3(9) 3(10) 135k

k

DefinitionsDefinitions

10

5

3 3(5) 3(6) 3(7) 3(8) 3(9) 3(10) 135i

i

Summand

Index of Summation

Limits of Summation

ExampleExample

1 2 3 4 55

1

1 1 1 1 1 1

2 2 2 2 2 2

1 1 1 1 1 11

2 4 8 16 32 32

k

k

Sigma Notation Representing Sigma Notation Representing Infinite SeriesInfinite Series

0 1 2 4

0

1 1 1 1 1...

2 2 2 2 2

1 1 11 ... 2

2 4 8

j

j

Give the series in expanded Give the series in expanded form: form:

5+10+15+205+10+15+20

4

1

5k

k

Find the Sum of Find the Sum of

190190

82

4i

i

Calculator steps: in LIST

One More: Find the Sum of One More: Find the Sum of

10891089

6

2

3k

k

Properties of SumsProperties of Sums

1 1 1

1 1 1 1 1 1

1. , constant 2. , constant

3. ( ) 4. ( )

n n n

i ii i i

n n n n n n

i i i i i i i ii i i i i i

c cn c is a ca c a c is a

a b a b a b a b

Last ProblemLast Problem

Find the sum of Find the sum of

4

0

( 1)

!

k

k k

3

8