8.1sequencesand summation.ppt

8/13/2019 8.1Sequencesand Summation.ppt

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Sequences & SummationSequences & Summation

NotationNotation8.18.1

JMerrill, 2007JMerrill, 2007

Revised 2008Revised 2008


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Sequences In Elementar Sc!ool"Sequences In Elementar Sc!ool"

12

12

#2


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$nd"$nd"

17

12


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EvenEven

12

22


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SequencesSequences

SEQUENCESEQUENCE % a set o num'ers, called% a set o num'ers, called

terms, arran(ed in a )articular order.terms, arran(ed in a )articular order.


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SequencesSequences

$n$n infiniteinfinitesequence is a unction *!osesequence is a unction *!ose

domain is t!e set o )ositive inte(ers. +!edomain is t!e set o )ositive inte(ers. +!e

unction values aunction values a11, a, a

22, a, a

##, ", a, ", a

nn" are t!e" are t!e

terms o t!e sequence.terms o t!e sequence.

I t!e domain o t!e sequence consists oI t!e domain o t!e sequence consists o

t!e irstt!e irst nn)ositive inte(ers onl, t!e)ositive inte(ers onl, t!e

sequence is asequence is a finitefinitesequence.sequence.

n is t!e term num'er.


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Eam)lesEam)les

-inite sequence-inite sequence 2, /, 10, 12, /, 10, 1

Ininite sequenceIninite sequence

1 1 1 1

, , , ,...2 4 8 16


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ritin( t!e +erms o a Sequenceritin( t!e +erms o a Sequence

rite t!e irst terms o t!e sequencerite t!e irst terms o t!e sequence

a ann #n 3 2 #n 3 2

aa11 #415 3 2 1 #415 3 2 1aa

22 #425 3 2 #425 3 2

aa## #4#5 3 2 7 #4#5 3 2 7

aa #45 3 2 10 #45 3 2 10

6alculator ste)s in IS+


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ritin( t!e +erms o a Sequenceritin( t!e +erms o a Sequence


a ann # 4%15 # 4%15nn

aa11 # 4%15 # 4%1511

22aa

22 # 4%15 # 4%1522

aa## # 4%15 # 4%15## 22

aa # 4%15 # 4%15


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9ou :o9ou :o


n

n

( 1)

a 2n 1

=

1 1 1 11, , , ,3 5 7 9


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;ra)!s;ra)!s

6onsider t!e ininite6onsider t!e ininitesequencesequence

5"41, = 5, 42, > 5"

1 1 1 1 1, , , , ..., ...2 4 8 16 2

n


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-indin( t!e n-indin( t!e nt!t!+erm o a Sequence+erm o a Sequence

rite an e)ression or t!e nrite an e)ression or t!e n t!t!term 4aterm 4ann5 o5 o

t!e sequence 1, #, ?, 7"t!e sequence 1, #, ?, 7"

n 1, 2, #, "nn 1, 2, #, "n

+erms 1, #, ?, 7"a+erms 1, #, ?, 7"ann

$))arent )attern eac! term is 1 less t!an$))arent )attern eac! term is 1 less t!ant*ice n. So, t!e a))arent nt*ice n. So, t!e a))arent n t!t!term isterm is

aann 2n % 1 2n % 1

$l*as com)are t!e term to

t!e term num'er


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-indin( t!e n-indin( t!e nt!t!+erm o a Sequence+erm o a Sequence

9ou :o9ou :o

rite an e)ression or t!e nrite an e)ression or t!e n t!t!term 4aterm 4ann5 o5 o

t!e sequencet!e sequence

$))arent )attern$))arent )attern+!e numerator is 1@ t!e denominator is t!e

square o n.

n 1, 2, #, "n1 1 1 11, , , , ...4 9 1 6 2 5 1 1 1 1

1, , , , ...4 9 1 6 2 5

nTerms a=

2

1na

n

=


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Recursive :einitionRecursive :einition

Sometimes a sequence is deined 'Sometimes a sequence is deined '(ivin( t!e value o a(ivin( t!e value o a

nnin terms o t!ein terms o t!e

)recedin( term, a)recedin( term, an-1n-1.. $ recursive sequence$ recursive sequence

consists o 2 )artsconsists o 2 )arts$n$n initial conditioninitial conditiont!at tells *!ere t!et!at tells *!ere t!esequence starts.sequence starts.

$$ recursive equationrecursive equation4or ormula5 t!at tells4or ormula5 t!at tells!o* man terms in t!e sequence are!o* man terms in t!e sequence arerelated to t!e )recedin( term.related to t!e )recedin( term.


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Eam)leEam)le

I aI ann aa

n-1n-1 and and aa

11 #, (ive t!e irst ive #, (ive t!e irst ive

terms o t!e sequence.terms o t!e sequence.

aa11 # #I n 2I n 2 aa

22 aa

11 # 7 # 7

I n #I n # aa33 aa

22 7 11 7 11

I n I n aa44 aa

33 11 1? 11 1?

I n ?I n ? aa5 aa44++ 1? 1A 1? 1A


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$ -amous Recursive Sequence$ -amous Recursive Sequence

+!e -i'onacci Sequence is ver *ell+!e -i'onacci Sequence is ver *ell

Bno*n 'ecause it a))ears in nature.Bno*n 'ecause it a))ears in nature.

+!e sequence is 1, 1, 2, #, ?, 8, 1#"+!e sequence is 1, 1, 2, #, ?, 8, 1#"

$))arent )atternC$))arent )atternC

Eac! term is t!e sum o t!e )recedin( 2Eac! term is t!e sum o t!e )recedin( 2

termsterms+!e nt! term is+!e nt! term is

aann a a

n%2n%2 a a

n%1n%1


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Eam)leEam)le


aa00 1 1

aa11 2 2

aa22 2 2aa

## D# D#

aa 2D# 2D#

n

n

2a , begin with n 0

n!= =


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-actorial Notation-actorial Notation

roducts o consecutive )ositive inte(ersroducts o consecutive )ositive inte(ersoccur quite oten in sequences. +!eseoccur quite oten in sequences. +!ese)roducts can 'e e)ressed in actorial)roducts can 'e e)ressed in actorial

notationnotation1F 11F 1

2F 22F 2 G 1 2G 1 2

#F # G2 G1 /#F # G2 G1 /F G# G2 G1 2F G# G2 G1 2

?F ? G G# G2 G1 120?F ? G G# G2 G1 120

+!e actorial Be can 'e

ound in M$+H R


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Eam)leEam)le

rite t!e irst our terms o t!e sequencerite t!e irst our terms o t!e sequence

n

n

2a

(n 1) !=

1

1

2

2

3

3

4

4

2 2 2a 2

(1 1)! 0! 12 4 4

a 4(2 1)! 1! 1

2 8 8a 4(3 1)! 2! 2

2 16 16 8a

(4 1)! 3! 6 3

= = = =

= = = =

= = = =

= = = =


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Evaluatin( -actorials in -ractionsEvaluatin( -actorials in -ractions

EvaluateEvaluate

( )n 1 !10!

2!8! n !

+

10 9 8! 9045

2 1 8! 2

= =

(n 1) n !n 1

n !

+ = +


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:einitions:einitions

+!e *ords sequences and series are oten+!e *ords sequences and series are oten

used interc!an(ea'l in everdaused interc!an(ea'l in everda

conversation. 4$ )erson ma reer to aconversation. 4$ )erson ma reer to a

sequence o events or a series o events.5sequence o events or a series o events.5In mat!ematics, t!e are ver dierent.In mat!ematics, t!e are ver dierent.

SequenceSequence a set o num'ers, terms,a set o num'ers, terms,

arran(ed in a )articular orderarran(ed in a )articular order

SeriesSeries t!e sum o a sequencet!e sum o a sequence


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Eam)lesEam)les

-inite sequence-inite sequence 2, /, 10, 12, /, 10, 1

-inite series-inite series 2 / 10 12 / 10 1

Ininite sequenceIninite sequence

Ininite seriesIninite series

1 1 1 1, , , ,...

2 4 8 16

1 1 1 1 ...2 4 8 16

+ + + +


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Intro to Si(maIntro to Si(ma

+!e ;reeB letter 4si(ma5 is oten used+!e ;reeB letter 4si(ma5 is oten used

in mat!ematics to re)resent a sumin mat!ematics to re)resent a sum

4series5 in a''reviated orm.4series5 in a''reviated orm.

Eam)leEam)le *!ic!*!ic!

can 'e read as t!e sum o Bcan 'e read as t!e sum o B 22or values oor values o

B rom 1 to 100.B rom 1 to 100.

1002 2 2 2 2

1

1 2 3 ... 100k

k

=

= + + + +


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:einition o a Series:einition o a Series

6onsider t!e ininite series a6onsider t!e ininite series a11, a, a

22, " a, " a

nn""

+!e sum o t!e irst n terms is a inite+!e sum o t!e irst n terms is a inite

series 4or )artial sum5 and is denoted 'series 4or )artial sum5 and is denoted '

+!e sum o all terms o an ininite+!e sum o all terms o an ininitesequence is called an ininite series and issequence is called an ininite series and is

denoted 'denoted '

1

n

i

i

a

=

1

i

i

a

=


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Si(ma 6ontinuedSi(ma 6ontinued

Similarl, t!e sm'ol is read t!eSimilarl, t!e sm'ol is read t!e

sum o #B or values o B rom ? to 10.sum o #B or values o B rom ? to 10.

+!is means t!at t!e sm'ol re)resents t!e+!is means t!at t!e sm'ol re)resents t!e

series *!ose terms are o'tained 'series *!ose terms are o'tained '

evaluatin( #B or B ?, B /, and so on, toevaluatin( #B or B ?, B /, and so on, toB 10.B 10.

10

5

3k

k

=

10

5

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

k

=

= + + + + + =


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:einitions:einitions

10

5

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

i=

= + + + + + =Summand

Inde o Summation

imits o Summation


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Eam)leEam)le

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

=

= + + + +

= + + =


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Si(ma Notation Re)resentin(Si(ma Notation Re)resentin(

Ininite SeriesIninite 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


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;ive t!e series in e)anded;ive t!e series in e)anded

ormorm

?101?20?101?20

4

1

5k

k=


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-ind t!e Sum o-ind t!e Sum o

1A01A0

82

4i

i=

6alculator ste)s in IS+


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Kne More -ind t!e Sum oKne More -ind t!e Sum o

108A108A

6

2

3k

k=


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ro)erties o Sumsro)erties o Sums

1 1 1

1 1 1 1 1 1

1. , constant 2. , constant

3. ( ) 4. ( )

n n n

i i

i i i

n n n n n n

i i i i i i i i

i i i i i i

c cn c is a ca c a c is a

a b a b a b a b

= = =

= = = = = =

= =

+ = + =


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ast ro'lemast ro'lem

-ind t!e sum o-ind t!e sum o 4

0

( 1)

!

k

k k=

3

8

8.1sequencesand summation.ppt

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